nanoparticle formation and dynamics in a complex (dusty) plasma

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Nanoparticle formation and dynamics in a complex(dusty) plasma : from the plasma ignition to the

afterglow.Lénaïc Couëdel

To cite this version:Lénaïc Couëdel. Nanoparticle formation and dynamics in a complex (dusty) plasma : from the plasmaignition to the afterglow.. Physics [physics]. Université d’Orléans, 2008. English. <tel-00349269>

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UNIVERSITÉ D’ORLÉANSécole doctorale sciences et technologies

laboratoire gremi

THE UNIVERSITY OF SYDNEYfaculty of science, school of physics

complex plasma laboratory

THÈSE/THESIS présentée par/presented by :Lénaïc Gaël Hervé Fabien COUËDEL

soutenue le /defended on : 11th septembre 2008pour obtenir les grades de/to obtain the titles of :

Docteur de l’Université d’OrléansDoctor of Philosophy of the University of Sydney

Spécialité/Specialty : Physique des Plasmas/Physics of Plasmas

Nanoparticle formation and dynamics in a complex (dusty) plasma :from the plasma ignition to the afterglow.

Formation et dynamique de nanoparticules dans un plasma complexe

(poussiéreux) : de l’allumage du plasma à la phase post-décharge.

THÈSE en COTUTELLE INTERNATIONALE dirigée par/supervised by :Monsieur Laïfa BOUFENDI Professeur, Université d’Orléans, FranceMonsieur Alexandre SAMARIAN Doctor, University of Sydney, Australia

RAPPORTEURS/EXAMINERS :Monsieur Gregor E. MORFILL Professor, Max Planck Institute, GermanyMonsieur Vladimir E. FORTOV Professor academician, Russian academy of science, Russia

JURY :Monsieur Jean-Michel POUVESLE Docteur, CNRS, France, Président du juryMonsieur Gregor E. MORFILL Professor, Max Planck Institute, GermanyMonsieur Vladimir E. FORTOV Professor academician, Russian academy of science, RussiaMonsieur Brian W. James Associate Professor, University of Sydney, AustraliaMonsieur Jean-Pierre BOEUF Docteur, CNRS, FranceMonsieur Christian STENZ Professeur, Université Bordeaux 1, FranceMonsieur Laïfa BOUFENDI Professeur, Université d’Orléans, FranceMonsieur Alexandre SAMARIAN Doctor, University of Sydney, Australia

Declaration of Originality

To the best of my knowledge, this thesis contains no copy or paraphrase of work published by

another person, except where duly acknowledged in the text. This thesis contains no material

which has been presented for a degree at the University of Sydney, the University of Orleans or

another university.

Lenaıc Couedel

1

Acknowledgments

Les travaux presentes dans cette these ont ete effectues au laboratoire GREMI (CNRS/Universite

d’Orleans, France) et au laboratoire CPL (School of Physics of the University of Sydney, Aus-

tralia). Il n’auraient pu etre menes a terme sans l’aide et le soutien de plusieurs personnes que

je souhaite remercier.

Je souhaite remercier Monsieur Jean-Michel Pouvesle, directeur du laboratoire GREMI, pour

avoir accepte de presider mon jury de these.

I would like to thank Prof. Gregor Morfill (Max Planck Institute, Garching, Germany) and

Prof. academician Vladimir Fortov (Russian academy of science) who accepted to be my thesis

examiners.

I would also like to thank A/Prof. Brian James (University of Sydney), Dr. Jean-Pierre Boeuf

(Laplace, Toulouse) and Prof. Christian Stenz (Universite Bordeaux 1) who accepted to be part

of my thesis jury.

Je remercie mes directeurs de these Prof Laıfa Boufendi (Universite d’Orleans) et Dr. Alexandre

Samarian (University of Sydney) qui ont rendu cette these en Cotutelle possible en facilitant la

collaboration entre mes deux laboratoires d’accueil ainsi que pour la qualite de leur encadrement.

Je souhaite particulierement remercier Dr. Maxime Mikikian pour son encadrement, ses compe-

tences, ses conseils et sa confiance durant toute la duree de ma these.

Je remercie egalement Dr. Pascal Brault pour ses conseils sur les techniques de simulation.

I would like to thank Dr. Joe Kachan (University of Sydney) and A/Prof. Brian James (again)

for their useful advices and corrections when I was writing my thesis manuscript.

Je souhaite remercier Yves Tessier, notre ingenieur au GREMI pour son aide sans faille durant

toute la duree de mon sejour au GREMI. Je voudrais aussi adresser un grand merci a Philippe,

Guy, Sebastien, Bernard, Jacky, Sylvie, Evelyne et Christophe pour leur aide precieuse durant

mon sejour au GREMI.

I would like to thank Robert Davies, our engineer in Sydney Uni who helped me with my ex-

periment and his faithful attendance to Monday soccer. I would like to give special thanks to

the Plasma group (Ian, Richard, Joe, Brian, Oded, Vanessa, Michael, John, Roberto, Alex) for

those very useful wednesday plasma group meetings.

Je veux aussi remercier tous les amis, collegues thesards et postdocs passes et presents du

GREMI ou d’ailleurs (Thomas, Marjorie, Xavier, Corinne, Gareth, Laurianne, Amael, Maria,

Thierry, Maxime, Alain, Clement, Gaetan, Elise, Fakhri, Edward, Sylvain, Binjie, Sebastien,

2

Herve, Jeremy, Catherine, Arnaud, Xeniya, ...pardon a ceux que j’oublie) grace a qui j’ai eu des

fous rires memorables et passe des soirees inoubliables. Merci Thomas, Edward et Amael pour

ces tournees “Rue de Bourgogne” (ainsi qu’ a Sydney et a Canberra pour le “Master of BBQs”)

permettant d’oublier un peu la these. Catherine, nous nous sommes connus que sur la fin de ma

these mais tu as une place speciale dans mon coeur.

I would like to thank all my friends in Sydney: Fitz, Katherine, John, Oded, Roberto, Mark,

Chanokporn, Alexey, Garth, Scott, Chris, Wojciech, Skander, the “physicist” soccer team,

Daren, Bill, Michael, Mike, Vanessa, Brett, Nick, Eve, Sebastien and all other mates attending

to the “Last Friday of the Month” drinks.

I would like to thank all my housemate in Sydney (Ollie, Kate, Laurie, Michelle, Luke, Char-

lotte, Bjorn) who survived in the same house as me. I had good fun with you and I hope we

will meet again someday.

I would like to thank Veronique for the “Paddy McGuires” tours and other parties in Sydney.

Un grand merci aux amis nantais et assimiles: Maxime, Nathalie, Pascal, Pierre, Tiburce, Yael,

Vincent, Fafane, etc.

I want also to give a special “thank you” to Marko and Heidi Lang with whom I became very

good friends during the past two years. I spent very good time with you and I will miss those

evenings at the Royal “living room” Hotel.

Je veux aussi adresser un grand merci a Christelle Monat et Christian Grillet pour toutes ces

apres-midis “Coinche” endiablees, Boston Legal, les soirees chansons francaises-cigares-whisky.

Grace a vous, je garde de merveilleux souvenirs de mon sejour a Sydney et je pense pouvoir dire

que nous sommes devenus de tres bons amis.

Enfin, je remercie ma famille pour leur soutien sans faille durant ces trois dernieres annees.

Mersi bras deoc’h

3

Contents

Declaration of Originality 1

Acknowledgements 2

Contents 7

List of figures 13

List of Constants 14

List of Symbols 18

List of Abbreviations 20

1 Introduction 21

1.1 A selective history of complex (dusty) plasma . . . . . . . . . . . . . . . . . . . . 21

1.1.1 The early age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.1.2 And it came from the sky . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.3 The industrial era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.1.4 Modern times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2.1 Aim of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.2.2 Organisation of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3 Introduction en Français . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Review of complex plasma physics 36

2.1 Charge and charge distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.1 The Debye length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.2 Isolated dust grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.3 Cloud of dust particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2 Forces acting on dust particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.1 The electric force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.2 The neutral drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4

CONTENTS

2.2.3 The ion drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.4 The gravitational force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.5 The thermophoretic force . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.6 Order of magnitude of the different forces . . . . . . . . . . . . . . . . . . 46

2.3 Instabilities and waves in complex plasmas . . . . . . . . . . . . . . . . . . . . . . 47

2.3.1 Characteristic frequencies of complex plasmas . . . . . . . . . . . . . . . . 47

2.3.2 Examples of waves and instabilities observed in complex plasmas . . . . . 48

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5 Résumé du !apitre en français . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Experimental Setups 51

3.1 Plasma chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 The PKE-Nefedov Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.2 The methane and silane reactor . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.3 The ComPLExS (Complex PLasma Experiment in Sydney) reactor . . . . 55

3.2 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Electrical diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.2 Optical and video diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.3 Laser extinction measurement . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.4 Emission spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.5 Electron density measurement by microwave resonant cavity . . . . . . . 64

3.2.6 Langmuir probe measurement . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2.7 Electron microscopy analysis . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72


4 Dust particle growth and instabilities 74

4.1 Dust particle growth in plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.1 Dust particle growth in Ar/CH4 plasma . . . . . . . . . . . . . . . . . . . 75

4.1.2 Dust particle growth in the PKE-Nefedov reactor . . . . . . . . . . . . . . 76

4.1.3 Dust particle growth in the ComPLExS reactor . . . . . . . . . . . . . . . 80

4.1.4 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.2 Dust particle growth instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Instabilities in the PKE-Nefedov chamber . . . . . . . . . . . . . . . . . . 88

4.2.2 Instabilities in ComPLExS . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100


5 Dust cloud instabilities in a complex plasma 103

5.1 The heartbeat instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.1 Classical heartbeats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5

CONTENTS

5.1.2 Failed or aborted heartbeat . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.1.3 Physical interpretation of the heartbeat instability . . . . . . . . . . . . . 123

5.2 Instabilities related to the growth of a new generation of dust particles . . . . . . 125

5.2.1 The delivery instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2.2 The rotating void . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132


6 Complex plasma afterglow 134

6.1 Afterglow in dust free plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.1.1 Diffusion in dust free plasma . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.1.2 Measurement of plasma decay in dust-free plasma . . . . . . . . . . . . . 139

6.2 Afterglow in complex plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.1 Diffusion in complex (dusty) plasma . . . . . . . . . . . . . . . . . . . . . 140

6.2.2 Measurement of plasma decay in complex plasmas . . . . . . . . . . . . . 145

6.2.3 Complex plasma afterglow as a diagnostics . . . . . . . . . . . . . . . . . 146

6.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3 Residual dust charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3.1 Evidence and measurement of residual dust charge . . . . . . . . . . . . . 149

6.3.2 Modelling of dust particle decharging . . . . . . . . . . . . . . . . . . . . 156

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170


7 General conclusion 172

7.1 General conclusion in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.2 Conclusion générale en français . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

A Dust charge distribution in running discharges 178

B EMD and HH spectrum 180

B.1 The empirical mode decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 181

B.2 The Hilbert Huang spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

B.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

B.3.1 Analysis of an analytic non-linear signal . . . . . . . . . . . . . . . . . . . 183

B.3.2 EMD of a heartbeat signal . . . . . . . . . . . . . . . . . . . . . . . . . . 185

C Program 188

C.1 Main program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

C.2 Subprogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

C.2.1 Initialisation of dust particle charges . . . . . . . . . . . . . . . . . . . . . 191

C.2.2 Characteristic times and coefficients . . . . . . . . . . . . . . . . . . . . . 192

6

CONTENTS

D Publications 197

D.1 Publications related to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 197



Bibliography 242

7

List of Figures

1.1 “spokes” on Saturn’s B ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 (a) Photograph of Laser Light Scattering during plasma operation from particle

structures suspended above a planar graphite electrode covered with three 82 mm

diameter Si wafers [32]. (b) SEM photo of a typical plasma-generated particulate

on a wafer surface after 12 hours exposure at a 200 mTorr, 10% CCl2F2 in Ar

plasma during Si etching [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1 Schematic of dust grains-particle j interaction . . . . . . . . . . . . . . . . . . . . 38

2.2 Calculation of dust particle charge . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 A dust particle immersed in a gas with a temperature gradient . . . . . . . . . . 46

3.1 Schematic of the PKE-Nefedov reactor . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Schematic of the Methane/Nitrogen(Argon) reactor . . . . . . . . . . . . . . . . . 54

3.3 ComPLExS reactor schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Principle schematic of optical diagnostics . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Laser extinction measurement schematic . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 Emission spectroscopy measurement schematic . . . . . . . . . . . . . . . . . . . 61

3.7 a) Line intensity ratio as a function of Te. b) Evolution of line relative intensity

as a function of Te . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.8 Principle schematic of microwave resonant cavity technique . . . . . . . . . . . . 66

3.9 Resonance curves of the microwave cavity with or without plasma . . . . . . . . 67

3.10 Typical I-V characteristic of a Langmuir probe . . . . . . . . . . . . . . . . . . . 68

3.11 Langmuir probe measurement schematic . . . . . . . . . . . . . . . . . . . . . . . 68

4.1 Evolution of the self-bias during the growth of dust particles in Ar/CH4 plasma 75

4.2 Dust particles grown in a Ar/CH4 plasma . . . . . . . . . . . . . . . . . . . . . . 76

4.3 a) Self-bias (Vdc) of the bottom electrode b)Amplitude of the fundamental har-

monics of the RF current (P=1.6 mbar and PW =3.25 W) . . . . . . . . . . . . . 77

4.4 Appearance time (a) as a function of pressure (b) as a function of the input power. 78

4.5 Current amplitude (blue curve) and relative laser intensity (black curve) as a

function of time during dust particle growth (P =1.6 mbar and PW =3.25 W). . 79

8

LIST OF FIGURES

4.6 a) Evolution of the intensity of the 750.39 nm argon line during particle growth

in the PKE-Nefedov reactor with pressure P = 1.6 mbar and input power PW =

3.25 W. b) Evolution of the Intensity ratio I763.5/I800.6 of argon lines during

particle growth for a pressure of P = 1.6 mbar and an input power PW = 3.3 W 80

4.7 Evolution of the self bias voltage of the powered electrode in the ComPLExS reac-

tor during dust particle growth (Pressure P=1.75 Torr and RF voltage Vpp=280 V) 81

4.8 Self-bias increase time as a function of: (left) pressure; (right) RF peak to peak

voltage at the start of the plasma run (Regimen B) . . . . . . . . . . . . . . . . 81

4.9 SEM images of the dust particles collected in the ComPLExS reactor for different

discharge times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.10 (a) Line intensities and (b) line ratio during dust particle growth at P = 1.75 Torr

and Vpp = 280 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.11 Evolution of the electron temperature Te estimated from the evolution of the line

ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.12 (a) Langmuir probe characteristic. The vertical arrow indicates the position of

the plasma potential. (b) First and second derivatives of the probe signal. (c)

Estimated EEDF from the second derivative of the probe characteristic (argon

plasma with P = 0.3 Torr and Vpp = 240 V). . . . . . . . . . . . . . . . . . . . . 85

4.13 I-V Langmuir probe characteristics at different instants of the discharge life (P =

1.75 Torr and Vpp = 280 V). The curves have been smoothed. . . . . . . . . . . 86

4.14 Evolution of plasma parameters obtained with Langmuir probe measurements in

an argon plasma with P = 1.75 Torr and Vpp = 280 V. (a) Evolution of the plasma

potential and the floating potential extracted from I-V curves. (b) Evolution of

the electron temperature deduced from the I-V curves. . . . . . . . . . . . . . . . 87

4.15 Current fundamental harmonic amplitude and corresponding spectrogram . . . . 89

4.16 Transition between the first 3 ordered phases (electric and optical signals . . . . 90

4.17 Fourier spectrogram and Hilbert-Huang spectrum during the three first phases

of DPGI. In this figure t = 0 does not correspond to the plasma ignition but

corresponds to a time close to the DPGI beginning. . . . . . . . . . . . . . . . . . 91

4.18 Duration of the P3 phase as a function of its frequency . . . . . . . . . . . . . . . 92

4.19 Fourier spectrograms of (a) electrical measurements (current fundamental har-

monic) and optical measurements (c) in the central fibre, (e) near the plasma

edge. (b) Zoom of (a), (d) zoom of (c), (f) zoom of (e) . . . . . . . . . . . . . . . 93

4.20 Evolution of the central line: (a) raw data extracts from the video (grey scale)

(b) Same as (a) but the mean time values of each pixel of the line have been

subtracted (in false colour from dark blue to dark red) . . . . . . . . . . . . . . . 94

4.21 Current signal measurements: (a) transition between P3 and P4 phases and en-

circles burst of order. (b) zoom of the encircled part of (a). . . . . . . . . . . . . 95

9

LIST OF FIGURES

4.22 Ordered domain during P4 phase: (a) raw line evolution from the video images.

(b) line evolution from the video images where the mean time values of each pixel

have been subtracted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.23 Fourier spectrum of the first chaotic regime. (a) on current signal (b) on central

optical fibre signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.24 Electrical and optical signal Fourier spectrum for DPGI . . . . . . . . . . . . . . 98

4.25 Growth instability appearance time as a function of P2 mean frequency . . . . . 98

4.26 Normalized current amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.27 Evolution of the self-bias voltage in the ComPLExS reactor for P=1.2 Torr and

Vpp =240 V. Instabilities during the growth appear as strong oscillations in the

self-bias voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.1 Void region in typical dust particle clouds with various sizes of grown particles

with crystalline regions and vortex motions (Image from Ref. [61]). . . . . . . . . 103

5.2 LRR heartbeat instability observed on the dust cloud. (a) Some images of the void

region during a CES. The blue ellipse delineates position of the stable open void.

(b) Time evolution of the central column profile (vertical line passing through

the void centre) constructed from the entire video (False colours from dark red

to bright yellow) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3 LRR heartbeat instability observed on the plasma glow (in false colour from dark

red to bright yellow). (a) Some images of the void region during a CES (image

354 is equivalent to image 30 in Fig.5.2). The blue ellipse delineates position of

the stable open void. (b) Intermediate images of (a) during fast plasma glow

modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 LRR heartbeat instability: current amplitude superimposed on column profile for

(a) only dust cloud; (b) dust cloud and plasma glow; (c) plasma glow only. Parts

(a), (b) and (c) are in false colours from dark blue to red . . . . . . . . . . . . . 108

5.5 Total plasma glow during a CES of LRR heartbeat instability: (a) some raw

images in grey scale directly extracted from the video; (b) same image as (a) with

post-processing and false colours (from dark red to bright yellow) . . . . . . . . . 109

5.6 LRR heartbeat instability: (a) Line profile (in false colours from dark blue to

red) extracted from the image series presented in Fig.5.5; (b) and (c) are different

magnifications of (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.7 LRR heartbeat instability: (a) Line profile (in false colours from dark blue to red)

calculated from the image series presented in Fig.5.5 with the current amplitude

superimposed; (b) zoom of a; (c) current amplitude; (d) central line (line 178)

profile of (a)); (e) middle position line (line 105) profile of (a) . . . . . . . . . . . 111

5.8 HRR heartbeat instability observed on the dust cloud: (a) some images of the

void region (in false colours from dark red to bright yellow; (b) time evolution of

the central column profile constructed from the entire video. . . . . . . . . . . . . 112

10

LIST OF FIGURES

5.9 HRR heartbeat instability observed on the total plasma glow: (a) some raw

images directly extracted from the video during a CES; (b) same images as (a)

with post processing and false colour (from dark red to bright yellow) . . . . . . 113

5.10 (a) Line profile (in false colour from dark blue to red) calculated from the image

series presented in Fig.5.9. (b) Zoom of (a) with electrical measurement super-

imposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.11 Extracted data from Fig.5.10: (a) amplitude of the current; (b) central line pro-

file; (c) line profile in the plasma edge (glow reversal region); (d) Plasma glow

integrated over a small central volume from complete image series of Fig.5.9 . . . 114

5.12 (a) Evolution of the central line luminosity of the images extract from very high

speed video (10,000 fps) during a HRR heartbeat (in false colour from blue to

red). (b) zoom of (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.13 Simultaneous record of the plasma glow and the dust cloud in the void region

during the setting up of the heartbeat instability. From image 135: stable open

void (delineated by a red ellipse) followed by a first contraction. A new contraction

starts at image 241 before the original conditions have been restored . . . . . . . 116

5.14 (a) Column profile (in false colour from dark red to bright yellow) extracted from

image series presented in Fig.5.13 with central plasma glow evolution superim-

posed (blue curve corresponding to line 95 of Fig.5.14(a)) (b) zoom of (a), (c)

corresponding line profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.15 Current during the heartbeat instability for different number of aborted peaks . 118

5.16 Transition from one aborted peak to two aborted peaks . . . . . . . . . . . . . . 120

5.17 Evolution of glow during a heartbeat with two failed peaks . . . . . . . . . . . . 121

5.18 Evolution of the central column extracted from the dust cloud video during a

heartbeat instability with two failed peaks. The dashed blue line superimposed

to the images represents the evolution of the current amplitude. . . . . . . . . . . 123

5.19 Dust cloud used for the study of the “delivery” instability. . . . . . . . . . . . . . 126

5.20 Evolution of the void during one expansion-contraction sequence of the delivery

instability. The blue ellipse delineates position of the stable open void. . . . . . . 127

5.21 a) Evolution of the central column going through the void (frame rate: 500 fps). b)

Zoom of a) with enhance contrast. The plasma glow luminosity was not totally

stopped by the interference filter allow us to visualize major glow luminosity

fluctuation at the same time as the dust cloud. . . . . . . . . . . . . . . . . . . . 127

5.22 (a) Evolution of the delivery instability frequency. The dashed curves represent

the theoretical fit. (b) Evolution of the delivery instability amplitude (1 pixel

∼ 100 µm). The dashed curves represent the theoretical fit. . . . . . . . . . . . 128

5.23 Evolution of the glow during one expansion-contraction sequence of a “delivery”

instability (in false colour from blue to red). In order to improve the visibility of

the luminosity variations, the image 1 is taken as the reference image and subtract

to every other images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

11

LIST OF FIGURES

5.24 Evolution of glow luminosity extracted from the video images presented in Fig.5.23

(in false colour from dark red to bright yellow). (a) Evolution of the central col-

umn luminosity. (b)Evolution of the central line luminosity (frame rate: 1789

fps). In both cases, the time average value of each pixels has been subtracted in

order to enhance luminosity variations. The dashed green line superimposed on

each picture represents the discharge current (AC component) during the delivery

instability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.25 (a) Rotating void: the void is not in the centre of the discharge and can be

observed on the right or on the left of the laser sheet alternatively. (b) Line

profile during a rotating void instability (in false colour from dark red to bright

yellow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.1 Evolution of the electron and ion diffusion coefficient (Ds; s = i, e) as a function

of (Λ/λDe)2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2 Evolution of resonance frequencies and relative electron densities in Argon and

nitrogen afterglow plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.3 Particle absorption time as a function of the dust density. a) With or without

taking into account the dust density effect. b) For different Te/Ti . . . . . . . . . 143

6.4 (a) Ratio of the dust particle absorption frequency taking into effect dust density

effect τ−1A relative to the dust particle absorption frequency not taking into effect

dust density effect τ−1A0

for different values of dust and plasma densities. (b)

Havnes parameter PH for different values of dust and plasma densities. . . . . . . 144

6.5 (a) Evolution of ion density ni during the first microsecond of a plasma afterglow

for different dust densities. (b) Evolution of the dust particle absorption frequency

τ−1A during the first microseconds of a plasma afterglow for different dust densities.

The diffusion frequency is τ−1D = 500 Hz . . . . . . . . . . . . . . . . . . . . . . . 144

6.6 Electron density decay in dusty and dust-free plasmas. . . . . . . . . . . . . . . . 145

6.7 Fit of experimental results of Ref.[21] using the first order approximation (Eq.6.44)147

6.8 Line profile of a high speed video imaging during the fall of the dust particle cloud

in discharge afterglow at P = 1.6 mbar (in false colour from blue to red). (a) No

bias on the lower electrode. (b) -10V bias on the powered electrode. (c) +10 V

bias on the power electrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.9 Schematic of the experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . 150

6.10 Thermophoresis in finite volume of gas . . . . . . . . . . . . . . . . . . . . . . . . 150

6.11 Temperature profile and gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.12 Superimposition of video frames taken with a large field of view CCD camera . . 151

6.13 Induced dust particle oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.14 Measured residual dust charge distributions. From the left to the right: P =

1.2 mbar with grown dust particles; P = 0.7 mbar with injected rd = 250 nm

dust particles; P = 0.4 mbar with grown dust particles. . . . . . . . . . . . . . . 155

12

LIST OF FIGURES

6.15 Qualitative time evolution of dust charge, plasma density and electron tempera-

ture during the afterglow. Four stages of the dust plasma decay can be identified:

I - temperature relaxation stage up to tT II - plasma density decay stage up tp,

III- dust charge volume stage tc, IV - frozen stage. . . . . . . . . . . . . . . . . . 158

6.16 Qualitative time evolution of the Havnes parameter after the power is switched off.160

6.17 Dust charge distribution for 190 nm radius dust particles with ni0 = 5 · 109 cm−3

and nd = 5 · 104 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.18 Decay of an argon plasma at P = 1.2 mbar with a fast ambipolar-to-free diffusion

transition. a) Electron temperature relaxation; b) density evolution; c) evolution

of diffusion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.19 Numerical results for 190 nm radius dust particles with argon pressure P =

0.4 mbar (P = 0.3 Torr) and nd = 5 · 104 cm−3. a) Ambipolar diffusion until the

end of the decay process. b) Abrupt transition from ambipolar to free diffusion

when PH = 0.5. c) Using data from Gerber and Gerardo for transition from

ambipolar to free diffusion [156]. d) Using data from Freiberg and Weaver for

transition from ambipolar to free diffusion [128] . . . . . . . . . . . . . . . . . . . 165

6.20 Numerical results for 190 nm radius dust particles with argon pressure P =

1.2 mbar (P = 0.9 Torr) and nd = 5 · 104 cm−3.a) Ambipolar diffusion until the

end of the decay process. b) Abrupt transition from ambipolar to free diffusion

when PH = 0.5. c) Using data from Gerber and Gerardo for transition from

ambipolar to free diffusion [156]. d) Using data from Freiberg and Weaver for

transition from ambipolar to free diffusion [128] . . . . . . . . . . . . . . . . . . . 166

6.21 Simulated DCDs at different times in the afterglow plasma (P = 1.2 mbar and∑

Nd = 1500). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.22 Evolution of nidiff/niabs

using data from Gerber and Gerardo for transition from

ambipolar to free diffusion [156]. a) P = 0.4 mbar. b) P = 1.2 mbar . . . . . . . 168

6.23 Evolution of |Qdnd/eni|. a) P = 0.4 mbar. b) P = 1.2 mbar . . . . . . . . . . . . 169

B.1 Non-linear signal of a wave with small frequency modulation and pure cosine wave

with the same mean frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

B.2 Fourier spectrogram and Hilbert-Huang spectrum of a non-linear wave signal with

small frequency modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

B.3 Evolution of the frequency of the non-linear wave . . . . . . . . . . . . . . . . . . 185

B.4 Heartbeat signal (AC current) and its IMFs from EMD . . . . . . . . . . . . . . 186

B.5 Heartbeat signal (solid line) and the sum of its largest amplitude IMFs (dashed-

line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

B.6 HHS and marginal Hilbert spectrum of the heartbeat signal . . . . . . . . . . . . 187

13

List of Constants

ǫ0 8.85419·10−12 A· s·V−1·m−1

π 3.14159265

c 2.99792458·108 m·s−1

e 1.60218·10−19 C

h 6.626076·10−34 J·skB 1.38066·10−23 J·K−1

14

List of Symbols

A probe area

Akj transition probability for k → j (j < k) radiative transition

Ar argon

C dust particle capacitance

CH4 methane

Cc dust particle charge corrective factor

Cabs absorption coefficient

Cext extinction parameter

Csca scattering coefficient

Da ambipolar diffusion coefficient

De electron diffusion coefficient

Di ion diffusion coefficient

Deff effective electron diffusion coefficient

E electric field

FT thermophoretic force

Fe electric force

Fdi ion drag force

Fdn neutral drag force

Fg gravitational force

He helium

Ia current absorbed by the dust particle

Ie electron current

Ii ion current

Ij j−species current

Il current emitted by the dust particle

It transmitted laser light

Ikj power radiated per unit of volume for k → j radiative transition

N number of bounded electrons

N2 nitrogen

15

List of Symbols

Nd number of dust particles

P pressure

PH Havnes parameter

PW RF power

PAr argon pressure

PHe helium pressure

Q quality factor of the microwave resonant cavity

Qd dust particle charge

Qdres residual dust particle charge

Qmean mean dust particle charge

Se electron source term

Si ion source term

SiH4 silane

T neutral gas temperature

Te electron temperature

Ti ion temperature

Teff effective electron temperature

V probe potential

Vf floating potential

Vp plasma potential

Vs sheath-presheath edge potential

VRF RF voltage

Vdc self-bias voltage

Vpp peak to peak RF voltage

W probability for a dust particle to carry the charge Qd

Z atomic number

Zd dust charge number

Zeff effective atomic number

Γ flux

Γc Coulomb logarithm

Γe electron flux

Γi ion flux

Λ diffusion length

αac accommodation coefficient

δ dust charge distribution width parameter

ǫr cavity dielectric constant

λD linearised Debye length

16

List of Symbols

λDe electron Debye length

λDi ion Debye length

λmfp electron or ion-neutral collisional mean free path

λrad radiation wavelength

µe electron mobility

µi ion mobility

µr cavity relative permeability

νe electron collision frequency

νen electron-neutral collision frequency

νkj radiation frequency between levels k and j

ω angular frequency

ωpd dust angular frequency

ωpe electron angular frequency

ωpi ion angular frequency

φd dust surface potential with respect to the plasma potential

φg dust surface potential

ρ mass density

σ standard deviation of the dust charge distribution

σdj cross section for a j particle-dust grain collision

σnd dust-neutral collision cross section

τA dust particle absorption time

τD diffusion time

τL plasma decay time

τQ charge fluctuation time

τT electron temperature relaxation time

τDe electron diffusion time

τDi ion diffusion time

Te Te/Ti

n ni,e/n0

ϕ dimensionless dust particle surface potential

bc OML maximum impact parameter

bj j-species impact parameter

bπ/2 impact parameter for 90 deflection

f frequency

fe electron velocity distribution function

fj j-species velocity distribution function

f1k absorption oscillator strength of the transition 1 → k

17

List of Symbols

ge electron energy distribution function

k1k electron impact excitation rate coefficient from the ground state

ktr translational part of the thermal conductivity

len electron-neutral collision mean free path

lin ion-neutral collision mean free path

m complex refractive index

me electron mass

mi ion mass

mj j-species mass

mn neutral atom mass

n cavity refractive index

n0 plasma density

n1 ground state density

nd dust particle density

ne electron density

ni ion density

nj j-species density

pe probability per unit of time for absorbing an electron

pi probability per unit of time for absorbing an ion

ptot probability per unit of time for absorbing a plasma particle

qj j-species charge

rd dust particle radius

v velocity

vd dust particle velocity

vi ion drift velocity

vs mean velocity of the ions approaching a dust particle

vTe electron thermal speed

vT i ion thermal speed

vTn neutral thermal speed

18

List of Abbreviations

ABR Allen-Boyd-Reynolds

ComPLExS Complex PLasma Experiment in Sydney

DA dust acousticDC direct currentDCD Dust charge distributionDIA dust-ion acousticDPGI dust particle growth instabilities

EEDF electron energy distribution functionEMD empirical mode decomposition

fps frames per second

GREMI Groupe de Recherches sur l’Energetique des Milieux Ionises

HH Hilbert-HuangHRR high repetition rate

IHED Institute for High Energy DensityIMF intrinsic mode functionISS International Space Station

LRR low repetition rate

MMO mixed-mode oscillationMPE Max Planck Institute for Extraterrestrial Physics

OML orbital motion limited

PKE Plasma Kristall Experiment

RF Radio-frequency

SEM scanning electron microscopy

TEM transmission electron microscopy

19

« La science consiste a passer d’un etonnement a un autre. »

Aristote

« The most exciting phrase to hear in science, the one that heralds new discoveries, is not

“Eureka”, but “That’s funny”. . . »

Isaac Asimov

20

Chapter 1

Introduction

Smoke, fog, soot, mist, dust all refer to particles that can be suspended in a liquid or a gas. Dusty

media have always been of considerable interest. The Egyptians used dust particles suspended

in a liquid to make ink, the Chinese discovered their explosive properties. Dust particles are

still of major interest in a wide range of applications from dry powder coatings [1], atmospheric

research [2, 3], to the microelectronics industry [4, 5]. In any cases, these particles can be useful

or harmful and consequently a lot of studies are devoted to them.

Dusty (or complex) plasmas are ionised gases in which there are some charged macroscopic dust

particles. They are very common media and can be found in nature such as in an astrophysical

environment (planetary nebula, planetary rings [6–8]) or planetary upper atmosphere [3, 9] as

well as in human made plasma (laboratory or industrial discharges [4], fusion plasmas [10, 11],

etc).

1.1 A selective history of complex (dusty) plasma

1.1.1 The early age

Before the XXth century, dusty plasmas were known as the zodiacal light (described and in-

terpreted by Cassini in the late 17th century) and the noctilucent clouds (discovered about a

century ago). However, the physics of dusty plasmas did not start before the observation of dust

particles in a laboratory discharge (hot cathode cylindrical positive column) by Irvin Langmuir

in the 1920’s [12]. Constriction and instabilities of the plasma were observed and assigned to the

presence of small dust particles coming from the sputtered tungsten cathode. From this time to

the 1960’s, the effect of dust and/or aggregates in low pressure plasmas was sporadically studied

(see for examples [13]).

The effect of dust particles in combustion plasmas (such as combustion products of solid rockets,

combustion products of hydrocarbon flames in which carbon particles may have formed, wake of

a re-entry body) was studied in the 1950’s and 1960’s (see for example Ref. [14] and references

therein). In most of these cases, the dust particles emit electrons thermionically (due a high

21

1.1 CHAPTER 1. INTRODUCTION

dust particle surface temperature), and therefore, their effects on the plasma are different from

the non-emiting dust particles that can be found in most low pressure and low temperature

plasmas.

In the late 1960’s, the effect of dust on DC positive columns gained attention again. The effect

of negatively charged large dust particles in a DC positive column was investigated. It was found

that the dust particle effect can be compared to the effect of negative ions in electronegative

discharges [15–19]. These results can be summarised as followed:

• When the dust density is not too high and when each dust particle can be considered

as isolated, electron and ion densities (ne and ni respectively) can be deduced from the

basic plasma equations [20] in which a term corresponding to volume losses on dust particle

surfaces is added (dust particles act as surface ionization sinks [17]). Moreover, the electron

temperature and the axial electric field were found to be higher than in a dust-free column.

• When the dust density is high, the dust particles replace the walls of the discharge and

the plasma can be roughly treated as a set of parallel plasma filaments between absorbing

walls with a radius about the order of the half of the mean interparticle distance [19].

• The dust particle surface temperature will be greater than that of the gas due to its heating

by surface recombination.

• The radial distribution of dust particles varies from one discharge to another and is a

compromise between a Maxwell-Boltzmann distribution, the effect due to local or large

scale gas movement, the chemical destruction of the particle and the ion drag force in a

strongly inhomogeneous discharge.

At the same time, Dimoff and Smy reported independantly that the dust particles act to

remove free electrons in low temperature plasmas [21]: they showed experimentally that the rate

of de-ionisation can be increased by the presence of dust particles. This result was obtained by

measuring the plasma decay time in the afterglow of a linear pulsed discharge. Nevertheless,

the theoretical explanation they provided was not satisfactory.

Meanwhile, the use of dust particles as a plasma diagnostic was also proposed [22, 23]:

• In Ref.[22], dust particles were observed between the capillary discharge and the Brewster

window during investigations on a He-Ne laser. It was proposed to use these dust particles

to locate potential local maxima in the discharge.

• In Ref.[23], it was proposed to inject dust particles inside a plasma and then, by measuring

their charges, to deduce the electron temperature. The injected dust particles were uniform

spherical alcohol droplets; then, by measuring the deflection of the dust particles in an

electric field, the charge carried by the dust particles was measured and the electron

temperature deduced.

22


1.1.2 And it came from the sky . . .

The studies of plasma-dust interactions are of major interest for astrophysics as dusty plasmas

can be found almost everywhere in a space environment (comet tails, planetary nebulae, planet

atmospheres, planetary rings) and their physical properties need to be understood to explain

observations.

Thus, a boost to the physics of dusty plasma came from astrophysics with the discovery of

Figure 1.1: “spokes” on Saturn’s B ring

nearly radial ghostly-like “spokes” in the outer portion of Saturn’s B ring in the early 1980’s

by the Voyager space probes (Fig.1.1). It brought forward a lot of theoretical works about

plasma-dust interactions in astrophysical environments. It was proposed that the spokes might

be dust particles charged to a sufficiently large negative potential, through electron collection

in transient dense plasmas, that they lift off from the ring and enter the permanently present

background plasma environment. The sufficiently dense short-lived transient plasma can be

created immediately above the ring by meteorite impacts [24, 25] or high energy auroral electron

beams [26]. The spokes are then sculpted by electrostatic forces. This theory led to a large

number of studies about the dust particle-plasma interactions and the charging processes of

dust particles in space plasmas (see for example Refs.[27, 28]).

However, the spokes in Saturn’s ring appear intermittently. Indeed, the spokes disappeared

from October 1998 to September 2005 when the Cassini spacecraft saw them reappear [29]. It is

suggested that, because the rings were more open to the sun than during Voyager missions and,

because the spokes depend critically on the background plasma density above the rings which

is a function of the solar angle, the charging environment prevented the formation of spokes.

Recently, it has been proposed that, due their localised formation, their seasonality and their

morphology, the “spokes are caused by lightning-induced electron beams striking the rings, at

locations magnetically-connected to thunderstorms” [30].

23


1.1.3 The industrial era

In the late 1980’s, dust particle formation was observed in laboratory and industrial chemically

active discharges (Fig.1.2) [4, 31]. Dust particle formation in an industrial reactor can be a sig-

nificant problem: dust particles falling on wafers (when the discharge is turned off) can adversely

affect the performance of semiconductor devices or the quality of thin films. Consequently the

mechanism of growth of dust particles in plasmas needed to be actively studied. It was indeed

a major issue to understand how dust particles were formed in the plasma and how to inhibit

their growth in order to avoid problems during industrial processes.

For this purpose, a lot of works has been carried out to understand the dust growth mechanism

(a) (b)

Figure 1.2: (a) Photograph of Laser Light Scattering during plasma operation from particlestructures suspended above a planar graphite electrode covered with three 82 mm diameter Siwafers [32]. (b) SEM photo of a typical plasma-generated particulate on a wafer surface after12 hours exposure at a 200 mTorr, 10% CCl2F2 in Ar plasma during Si etching [4].

in silane discharges which are commonly used in the industry (see Ref.[5] and references therein).

In Ref.[33–41], the dust particle growth mechanism and the dust cloud behaviour were studied in

argon-silane discharges. It was shown that dust growth occurs following a well defined pattern:

1. Formation of molecular precursors from gas dissociation

2. Formation and accumulation of nanocrystallites from these precursors

3. Aggregation of the nanocrystallites

4. Growth by molecular sticking

The transport of dust particles in the discharge has also been well studied (see for example

Refs.[10, 42]). Since the dust particles are generally charged in plasmas, they are subject to

different forces (ion drag force, neutral drag force, electrostatic force, gravity, thermophoresis)

which can confine or expel the particles from the discharge. Consequently it is possible to

manipulate the dust particles, which can have strong industrial applications [5].

24


1.1.4 Modern times

The additional dust component of complex plasmas offers a unique opportunity to study strongly

coupled systems as the coupling parameter Γc (representing the ratio of the potential electro-

static energy over the dust thermal energy) can be very large. It has indeed been shown that the

dust particle density can reach very high values and that the total charge carried by the dust

particles can exceed the one carried by the electrons. The coupling parameter Γc can be much

greater than one and the dust particle cloud can be considered as a Coulomb liquid [33, 38] or

even as Coulomb crystals [43–45].

As the dust particles interact strongly with each other, complex plasmas can be seen as self-

organising systems. Dust plasma crystals have been discovered in the near sheath areas of capac-

itively coupled radio-frequency (RF) discharges [43–45]. The discovery of dust plasma crystals

gave a real boost to the investigations of dusty plasmas. Many experiments were designed to

study complex plasmas with an added micrometer-sized dust component and the formation of

ordered structures were reported in DC discharges [46], Q-machines [47], thermal plasmas op-

erated at atmospheric pressures [48, 49], UV-induced plasmas [50] and nuclear-induced plasmas

[51].

Many studies (experimental and theoretical) are now devoted to the investigation of the dust

charging process, dust-dust interactions, dust-plasma interactions and the forces acting on the

dust particles (ion drag force, thermophoresis, neutral drag force, etc). For example, complex

plasmas allow the studies of phenomena such as wave propagation [52, 53], crystal melting and

phase transitions [54–56], and cluster motion to the kinetic level [57]. Formation of intriguing

dust structures such as the dust ”void” (the region totally free of dust in the core of the plasma)

and its instabilities have been reported under microgravity conditions [58] and in discharges

where high densities of grown particles can be achieved [59, 60].

The recent advances of experimental research under microgravity gave new opportunities to

study the properties of complex plasmas [61–64]. Indeed, under typical laboratory conditions,

the dust particles usually levitate in the sheath region where the electric field is strong enough

to balance the gravitational force and gravity thus limits the range of parameters that can be

studied. However, under microgravity conditions, a much wider range of parameters can be

studied and bigger dust particle sizes can be used.

Experiments under microgravity conditions were first considered in the beginning of the 90’s.

Experiments were done at the end of the 90’s on board parabolic flights, with rocket launches

(TEXUS) and on board the MIR space station [65], mainly by two laboratories: the Max Planck

Institute for Extraterrestrial Physics (MPE) in Garching (Germany) and the Institute for High

Energy Density (IHED) in Moscow (Russia). These laboratories decided at the end of 1998 to

collaborate through the program PKE (Plasma Kristall Experiment). PKE-Nefedov was the

first plasma experiment designed for use on board the International Space Station (ISS). It was

built to study dusty (complex) plasmas under microgravity conditions to avoid problems asso-

ciated with the weight of the dust particles in ground-based experiments. As a result, it was

possible to directly study plasma-dust particle interactions and dust particle-dust particle inter-

25


actions without gravitational perturbations. At the beginning of 2001, the first experiment with

the PKE-Nefedov apparatus was made on board the ISS. Micrometer size melamine formalde-

hyde dust particles were injected directly into the plasma in order to study phenomena such as

Coulomb crystals.

In December 2000, the GREMI laboratory (Orleans, France) joined the collaboration. The aim

was to study the growth of particles created by sputtering of the previously injected dust parti-

cles. These grown dust particles are much smaller than the injected ones and can fill the whole

chamber, even in ground-based experiments. At the end of 2001, the astronaut C. Haignere

made the first experiment of dust particle growth in the PKE-Nefedov reactor during the An-

dromeda mission. Some results obtained under microgravity conditions have been partially

reproduced in ground-based experiments [61, 66–68]. Since that time, dust particle growth in

the PKE-Nefedov reactor and phenomena such as growth instabilities and the void “heartbeat”

oscillations are under active investigation. Complex plasmas under microgravity conditions are

still intensively studied with the PK3+ experiment (an improved version of the PKE-Nefedov

experiment) on board the ISS [69], and the PK4 experiment (a DC discharge) is under active

development [70].

Meanwhile, the research on dust plasma interaction and particle growth is still motivated by

strong technological challenges. For example, dust particles created by plasma-wall interactions

in thermonuclear fusion devices are actively studied as the dust particles can be very harm-

ful (lowering of plasma temperature, safety, transport of impurities, radioactivity) and lead to

the malfunction of the fusion reactor [10, 11]. On the other hand, nanoparticles grown in the

plasma phase can have very interesting and useful properties. For example, the electronic proper-

ties of polymorphous silicon (amorphous silicon with embedded crystalline silicon nanoparticles)

created in silane-based discharges are of major interest for the manufacture of solar cells [71, 72].

1.2 Overview of this thesis

1.2.1 Aim of this thesis

Until now, the different studies on complex/dusty plasmas have focussed on few aspects of

the new physics resulting from the presence of dust in the plasma. However, it is obvious

that a complete picture from the plasma ignition to the discharge afterglow has to be built as

every industrial or laboratory discharge has to be turned on and turned off. Consequently, in

this thesis, phenomena arising from the growth of the dust particles at plasma ignition to the

discharge afterglow are investigated.

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1.2.2 Organisation of this thesis

Chapter 2 and chapter 3 are devoted to the basics of complex (dusty) plasma physics and the

description of the experiments. Then, this thesis is centred around three major issues:

1. Growth of dust particles by sputtering in capacitively coupled RF discharges and associated

instabilities (Chapter 4)

2. Instabilities of dust particle clouds in a complex plasma (Chapter 5)

3. Influence of dust particles on complex plasma decay and dust particle residual charges

(Chapter 6)

As can be seen, the main goal of this thesis is to give a complete picture of a complex plasma

from its beginning until it is extinguished. However, the work in each part of this thesis has

been motivated by previous results that needed to be further investigated:

1. The first issue (Chapter 4) is devoted to the dust particle growth by sputtering in a RF

discharge and its influence on the plasma and discharge parameters. This work was moti-

vated by the activities of Goree’s group [59, 73, 74] and is in the continuity of Mikikian’s

experiments at the GREMI laboratory [61, 66, 68, 75]. Its main goal is a better under-

standing of the effect of dust particle growth on the discharge and plasma parameters. For

this reason, dust particle growth by RF sputtering has been investigated in two different

RF discharges: the PKE-Nefedov reactor and the ComPLExS reactor. The diagnostics

that have been used are electrical measurements (self bias voltage and current amplitude),

optical measurements (high speed video imaging, spectroscopy) and Langmuir probe mea-

surements. It is reported that the self bias voltage decreases in absolute value during the

dust particle growth and that the current amplitude decreases at the same time. It is

attributed to the loss of free electrons to the growing dust particle surfaces. It is also

reported that the electron energy distribution function EEDF is greatly affected by the

growth of the dust particles (increase of the electron temperature but loss of the energetic

tail of the EEDF). It is also reported that the dust particle growth is greatly affected by

the gas purity in the discharge. Finally, dust particle growth instabilities (DPGI) have

been observed. It is shown that these instabilities follow a defined evolution pattern.

2. The second issue (Chapter 5) focusses on dust particle cloud instabilities in a complex

plasma. We focus on the instabilities of the void region. Our results are concerned with

the heartbeat instability (regular contractions and expansions of the void size) which has

been studied theoretically [58, 76] and experimentally [60, 61] and void instabilities due to

the growth of a new generation of dust particles inside the void: the “delivery instability”

(which is accompanied by low frequency and large amplitude contraction and expansion

of the void region and that ends with a new generation of dust particles inside the former

void region), and the rotating void (which is characterised by a rotation of the void region

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around the central axis of the chamber). In these studies, we focus particularly on mea-

surements of some discharge parameters (discharge current) as well as spatially resolved

measurements of the plasma glow (high speed video imaging, spatially focussed optical

fibres coupled to photomultiplier tube (PMT) array) and observations of the dust cloud

motion (high speed video imaging). Correlations between the different measurements led

to a better understanding of the physical phenomena involved in the instabilities of dusty

plasmas. It has been found that the dust void contraction is preceded by a strong increase

of the plasma glow inside the void and hence the ionisation. Moreover, failed or aborted

contraction can occur in between the main contraction-expansion sequences of the heart-

beat instability. The evolution kinetics of the delivery instability is found to be linked to

the growth kinetics of the new dust particle generation.

3. The third and last issue (Chapter 6) focusses on the study of dusty plasma afterglow and

the residual electric charge on the dust particles when the plasma has disappeared. This

work emerged as a result of the discovery of residual electric charge on dust particles after

the power of an RF discharge was switched off during microgravity experiments [77]. Be-

fore this result, it was commonly believed that dust particles do lose their electric charge

during the afterglow and remain neutral in the surrounding gas. Hence more investigations

were necessary.

It naturally led to the study of dusty afterglow plasma in order to understand the decharg-

ing process of the dust particles. Since the dust particle (de)charging process is strongly

linked to the diffusion of the charge carriers in the afterglow plasma, theoretical investi-

gations about the influence of the dust particle in the plasma loss processes in discharge

afterglow have been made. Comparison with the available data [21] shows that the mea-

surement of decay time in the afterglow plasma may be used as a complementary diag-

nostic. Furthermore, measurements of the electron density decay in various dust-free and

dusty afterglow plasmas have been performed. The results confirm that dust particles

influence substantially the diffusion of electrons in afterglow plasma.

In order to further investigate the dust residual charges in the late plasma afterglow,

ground-based measurements on dust residual charge in the PKE-Nefedov reactor have

been performed. The dust particles were directly grown in the plasma chamber by sput-

tering of polymer materials previously deposited on the electrodes. The dust particles

were sustained in the chamber volume after the discharge was switched off by a system

creating an upward thermophoretic force inspired by Rothermel’s experiments [78]. We

confirm that the dust particles do carry residual electric charges and we show that this

charge can be either positive or negative. Dust charge distributions (DCDs) have been

reconstructed from measurements of many dust particle charges and show that the width

of the dust charge distribution is larger than the one expected for the running discharge.

Simulations of the decharging process show that the residual dust charge distribution is

very sensitive to the ion and electron diffusion losses and especially the transition from

ambipolar to free diffusion.

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To conclude, in this thesis, we focus on the influence of dust particles on RF discharges from

the plasma ignition to the discharge afterglow. It allows us to perform a detailed analysis of the

effect the dust particles can have on a RF discharge at different instants during its lifetime. The

main results are centred around the different stages of the lifetime of the discharge:

1. The effect of dust particle growth at plasma ignition and its effect on the plasma and

discharge parameters and the related instabilities

2. The instabilities of the void region in a well established dust cloud in a discharge

3. The effect of dust particles on afterglow plasma and the influence of the plasma diffusion

losses on the particle decharging and the residual charges.

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1.3 Introduction en Français

La fumee, le brouillard, la suie, la brume, la poussiere sont autant de termes qui renvoient ades particules en suspension dans des liquides ou des gaz. Les milieux poussiereux ont toujoursete tres interessants pour l’homme. Les egyptiens ont utilise des poudres en suspension dans unliquide pour creer l’encre ; les chinois ont decouvert leurs proprietes explosives. Les poudres res-tent aujourd’hui interessantes pour une grande variete d’applications allant des enduits secs [1]aux recherches sur l’atmosphere [2, 3] en passant par l’industrie de la microelectronique [4, 5].Dans tous les cas, les poudres peuvent etre soit utiles, soit nuisibles et, par consequent, de nom-breuses etudes leur sont consacrees.Les plasmas poussiereux (ou complexes) sont des gaz ionises contenant des poudres macro-scopiques chargees. Ce sont des milieux communement rencontres dans l’espace (nebuleusesplanetaires, anneaux planetaires [6–8]), les hautes atmospheres [3, 9] et dans les plasmas creespar l’homme (decharges industrielles ou de laboratoire [4], plasmas de fusion [10, 11], etc).

Histoire sélective des plasmas poussiéreux

La préhistoire

A l’aube du XXeme siecle, les plasmas poussiereux n’etaient connus que sous la forme delueurs zodiacales (correctement decrites et interpretees par Cassini a la fin du XVIIeme siecle)et de nuages noctilucents (decouverts il y a environ un siecle). Leur etude physique n’a cependantpas commence avant l’observation de poudres dans une decharge de laboratoire (colonne positivecylindrique a cathode chaude) par Irvin Langmuir dans les annees 1920 [12]. Des constrictionset des instabilites du plasma furent observees et attribuees a la presence des petites poudresprovenant de la pulverisation de la cathode de tungstene. En revanche, jusqu’aux annees 1960,l’impact de la presence de poudres dans les plasmas basse pression ne fut que sporadiquementetudie (voir par exemple la Ref.[13]).L’effet des poudres dans les plasmas de combustion (comme les produits de combustion des fuseesa combustible solide, les produits de combustion de flammes d’hydrocarbures dans lesquelles dessuies de carbone peuvent etre crees, le sillage d’objets entrant dans l’atmosphere) fut etudie dansles annees 1950 et 1960 (voir par exemple la Ref.[14] et ses references). Dans la majorite deces etudes, les poudres sont emettrices d’electrons thermoioniques (a cause d’une importantetemperature de surface) et, par consequent, doivent avoir un impact different sur les plasmas decelles non-emettrices que l’on peut trouver dans la plupart des plasmas basse pression et bassetemperature.A la fin des annees 1960, l’influence des poudres sur les colonnes positives a courant continuconnut un regain d’attention. L’effet de poudres massives chargees negativement sur des colonnespositives a courant continu fut etudie. Il a ete montre qu’elles ont un effet comparable a celuides ions negatifs dans les decharges electronegatives [15–19]. Ces resultas peuvent etre resumesde la maniere suivante :

– Quand la densite de poussieres n’est pas trop importante et quand chaque poudre peut etreconsideree comme isolee, les densites d’ions et d’electrons (respectivement ni et ne) peuventetre deduites des equations plasmas classiques [20] dans lesquelles un terme correspondantaux pertes en volume dues aux recombinaisons a la surface des poudres a ete ajoute. Deplus, il a ete observe que la temperature electronique ainsi que le champ electrique axialsont plus importants que dans une colonne sans poudre.

– Quand la densite de poussieres est importante, les poudres remplacent les parois de ladecharge et le plasma peut etre approximativement vu comme un jeu de filaments de plasmaparalleles entre des parois absorbantes, chacun avec un rayon de l’ordre de la moitie de ladistance moyenne entre deux poudres.

– La temperature de surface des poudres est plus elevee que celle du gaz environnant a causede l’echauffement du aux recombinaisons de surface.

30


– La distribution radiale des poussieres varie d’une decharge a l’autre et est un compromisentre une distribution de Maxwell-Boltzmann, l’effet des mouvements locaux et a grandeechelle du gaz environnant, de la destruction chimique des poudres et de la force de frictiondes ions dans les decharges fortement inhomogenes.

A la meme epoque, Dimoff and Smy rapporterent independamment que les poudres ont ten-dance a capturer les electrons libres dans les plasmas basse temperature [21] : ils montrerentexperimentalement que le taux de de-ionisation peut etre accru par la presence des poussieres. Ceresultat fut obtenu en mesurant le temps de decroissance du plasma dans la phase post-decharged’une decharge lineaire pulsee. Leur explication theorique n’etait neanmoins pas satisfaisante.Pendant ce temps, l’utilisation de poudres comme moyen de diagnostic pour les plasmas a aussiete suggeree :

– Dans la Ref.[22], des poussieres furent observees entre la decharge capillaire et la fenetrede Brewster lors d’etudes sur les lasers He-Ne. Ils proposerent d’utiliser ces poussierespour localiser les maxima locaux de potentiel dans la decharge.

– Dans la Ref.[23], il fut propose de mesurer la temperature electronique d’un plasma en me-surant les charges electriques de particules injectees a l’interieur de celui-ci : les particulesinjectees etaient des gouttes d’alcool qui ont la propriete d’etre spheriques et uniformes ;ensuite, en mesurant la deflexion de leurs trajectoires dans un champ electrique, leurscharges electriques etaient mesurees et la temperature electronique deduite.

Et cela vint du ciel . . .

L’etude des interactions entre les poudres et les plasmas est d’un grand interet pour l’astro-physique. En effet, les plasmas poussiereux peuvent etre rencontres un peu partout dans l’espace(queues des cometes, nebuleuses planetaires, atmospheres planetaires, anneaux planetaires) etleurs proprietes physiques ont besoin d’etre comprises afin d’expliquer les observations.Ainsi, un coup d’accelerateur a la physique des plasmas poussiereux vint de l’astrophysique gracea la decouverte de « stries »(« spokes ») pratiquement radiales sur la partie exterieure de l’an-neau B de Saturne par la sonde Voyager au debut des annees 1980 (Fig.1.1). Cela a engendre ungrand nombre de travaux sur les interactions plasmas-poudres dans les milieux astrophysiques. Ilfut propose que ces stries soient composees de poussieres chargees a des potentiels negatifs suffi-samment importants, par collection d’electrons dans des plasmas transitoires denses, afin de leselever au dessus de l’anneau et de les faire entrer dans le plasma environnant. Des plasmas densestransitoires peuvent etre crees juste au dessus des anneaux lors d’impacts avec des meteorites[24, 25] ou par des faisceaux auroraux d’electrons de hautes energies [26]. Les stries sont ensuitesculptees par les forces electrostatiques. De nombreuses etudes furent ensuite consacrees aux in-teractions plasmas-poussieres et aux processus de chargement des poudres (voir par exemples lesRefs.[27, 28]).Les stries sur les anneaux de Saturne apparaissent cependant par intermittence. En effet, aucunestrie ne fut observee d’octobre 1998 a septembre 2005 au moment ou la sonde Cassini les a vuesreapparaıtre [29]. L’hypothese proposee est que, parce que les anneaux etaient plus exposes ausoleil que pendant les missions Voyager et parce que les stries sont tres dependantes de la den-site du plasma environnant, la formation des stries fut inhibee par les conditions ambiantes. Il arecemment ete suggere que, en raison de leur formation localisee, leur saisonnalite et leur mor-phologie, les stries sont generees par des faisceaux d’electrons induits par la foudre et atteignantles anneaux en des domaines connectes magnetiquement aux orages [30].

L’ère industrielle

A la fin des annees 1980, la formation de poussieres fut observee dans des decharges chimi-quement actives industrielles et de laboratoire (Fig.1.2) [4, 31]. La formation de poudres dans lesreacteurs industriels peut etre un gros probleme : les particules tombant sur les substrats peuventreduire la performance des circuits integres ou la qualite des couches minces. Par consequent,les mecanismes de croissance de poudres dans les plasmas devaient etre etudies finement. Il etait

31


en effet vital de comprendre comment les poussieres se forment dans les plasmas et commentinhiber leur formation afin d’eviter les problemes dans les processus industriels.Pour ces raisons, de nombreux travaux ont ete menes pour comprendre les mecanismes de crois-sance de poudres dans les decharges a base de silane tres largement utilisees dans l’industrie(voir la Ref.[5] et ses references). Dans les Refs.[33–41], la croissance et le comportement denuages de poudres dans des decharges argon-silane ont ete etudies. Il a ainsi ete montre que lacroissance des poussieres suit le modele suivant :

1. Precurseurs moleculaires provenant de la dissociation du gaz

2. Formation et accumulation de nanocristallites

3. Agglomeration des nanocristallites

4. Croissance par attachement moleculaire

Le transport des particules dans les decharges a aussi ete etudie (voir par exemple les Refs.[10,42]). Les poudres etant generalement chargees dans les plasmas, elles sont soumises a differentesforces (force de friction des ions, force de friction des neutres, forces electrostatiques, gravite,thermophorese) qui peuvent les confiner ou les expulser de la decharge. Il est par consequentpossible de manipuler les particles, ce qui peut avoir d’importantes applications industrielles [5].

Les temps modernes

La composante additionnelle de poudres chargees dans les plasmas poussiereux donne l’oppor-tunite unique d’etudier les systemes fortement couples en raison du parametre de couplage Γc (quirepresente le rapport entre l’energie potentielle electrostatique des poudres et leur energie d’agi-tation thermique) pouvant etre tres grand. Il a en effet ete rapporte que la densite de poussieresdans un plasma peut etre extremement elevee et que la charge globale portee par celles-ci peutdonc exceder celle portee par les electrons. La parametre de couplage Γc peut etre alors largementsuperieur a un et le nuage de poudres considere comme un liquide coulombien [33, 38], voirememe un cristal coulombien [43–45].En raison des fortes interactions des poudres entre elles, les plasmas poussiereux peuvent etrevus comme des milieux auto-organises. Des cristaux coulombiens furent ainsi decouverts dansles zones aux alentours des gaines dans des decharges radio-frequences (RF) capacitives [43–45]. Cette decouverte crea un reel engouement pour les recherches sur les plasmas poussiereux.De nombreuses experiences, dans lesquelles des poudres micrometriques etaient ajoutees, furentconcues pour ces travaux et la formation de structures ordonnees fut observee dans des dechargesa courant continu [46], des machines Q [47], des plasmas thermiques a pression atmospherique[48, 49], des plasmas induits par rayonnement ultraviolet [50], et des plasmas crees par desreactions nucleaires [51].De nombreuses etudes (experimentales et theoriques) sont dorenavant consacrees aux processusde chargement des poudres, aux interactions poudre-poudre, aux interactions poudre-plasma etaux forces agissant sur les poussieres (force de friction des ions, force de friction des neutres,thermophorese, etc). Les plasmas poussiereux permettent par exemple d’analyser certains pheno-menes comme la propagation d’ondes [52, 53], la fusion des cristaux et les transitions de phases[54–56], et la dynamique des clusters au niveau le plus elementaire [57]. La formation de struc-tures singulieres comme le « void »(region vide de poudres au centre du plasma) et ses instabilitesont ete observees lors d’experiences en microgravite [58] et dans des decharges ou la densite depoudres formees dans le plasma peut etre tres elevee [59, 60].Les recentes avancees aux niveaux des recherches en microgravite ont ouvert de nouvelles perspec-tives pour l’etude des proprietes des plasmas poussiereux [61–64]. En effet, durant les experiencesen laboratoire, les poudres levitent generalement dans les gaines du plasma ou le champ electriqueest suffisamment puissant pour compenser leur poids. L’eventail de parametres qui peuvent etreexplores est alors restreint. Au contraire, lors d’experiences en microgravite, la gamme peut etreetendue et des poudres plus grosses peuvent etre utilisees.Les experiences en microgravite ont d’abord ete envisagees au debut des annees 1990. Lespremieres furent realisees a la fin des annees 1990 a bord de vols paraboliques et de fusees, et

32


a bord de la station spatiale MIR par principalement deux laboratoires : le Max-Planck-Institutfur Extraterrestrische Physik (Allemagne) et l’Institut for High Energy Density (IHED, Russie).Ces laboratoires deciderent a la fin de l’annee 1998 de collaborer et creerent le programme PKE(Plasma Kristall Experiment). PKE-Nefedov fut ainsi la premiere experience plasma concue pourfonctionner a bord de la station spatiale internationale (ISS). Elle a ete construite afin d’etudierles plasmas poussiereux en microgravite et ainsi eviter les problemes dus au poids des poudres.Il fut ainsi possible d’etudier directement les interactions plasma-poudre et poudre-poudre sansles perturbations liees a la gravite. Au debut de l’annee 2001, les premieres experiences a bordde l’ISS furent realisees. Des poudres de melamine formaldehyde de quelques micrometres furentinjectees dans le plasma afin d’etudier les structures ordonnees comme les cristaux coulombiens.En decembre 2000, le laboratoire GREMI (France) joignit la collaboration afin d’etudier la crois-sance de poudres par pulverisation des poudres precedemment injectees dans le reacteur. Cespoussieres sont bien plus petites que les poudres injectees et peuvent remplir la decharge memeen laboratoire. Fin 2001, l’astronaute C. Haignere fit les premieres experiences sur la crois-sance de poudres dans le reacteur PKE-Nefedov durant la mission Andromede. De plus, certainsresultats obtenus en microgravite furent partiellement reproduits en laboratoire [61, 66–68]. De-puis cette epoque, la croissance de poudres dans le reacteur PKE-Nefedov et les phenomenescomme les instabilites de croissance et les battements du « void »sont intensivement etudies.Les plasmas poussiereux en microgravite sont toujours etudies grace a l’experience PK3+ (uneversion amelioree de PKE-Nefedov) a bord de l’ISS [69], et le developpement de l’experiencePK4 (une decharge a courant continu) [70].

Pendant ce temps, les recherches sur les plasmas poussiereux sont toujours motivees pard’importants defis technologiques. Par exemple, les poussieres creees lors des interactions duplasma avec les parois des tokamaks pour la fusion nucleaire sont fortement etudiees car ellespeuvent etre extremement nuisibles (reduction de la temperature du plasma, securite, transportd’impuretes, radioactivite) et entraıner un dysfonctionnement du reacteur [10, 11]. D’un autrecote, les nanopoudres formees dans les plasmas peuvent avoir des proprietes tres interessantes.Par exemple, les proprietes du silicium polymorphe (silicium amorphe incruste de nanocristauxde silicium cree dans des decharges a base de silane) sont tres interessantes pour la fabricationde panneaux solaires [71, 72].

Présentation de la thèse

But de la thèse

Jusqu’a maintenant, les differentes recherches sur les plasmas poussiereux se sont focaliseessur certains aspects de la nouvelle physique due a la presence des poudres. Cependant, il estevident qu’un panorama complet depuis l’allumage du plasma jusqu’a la phase post-dechargedoit etre construit car chaque decharge industrielle ou de laboratoire doit etre allumee maisaussi eteinte. Par consequent, dans cette these, les phenomenes lies a la croissance de poudresdans le plasma depuis l’allumage jusqu’a la phase post-decharge sont etudies.

Organisation de la thèse

Les chapitres 2 et 3 sont dedies aux bases de la physique des plasmas poussiereux et a ladescription des experiences. Ensuite, la these s’articule autour de trois grands themes :

1. Croissance de poudres par pulverisation dans des decharges RF capacitives et les instabilitesassociees (Chapitre 4).

2. Les instabilites des nuages de poussieres dans les plasmas complexes (Chapitre 5).

3. L’influence des poudres sur la decroissance du plasma durant la phase post-decharge et lescharges residuelles des poussieres (Chapitre 6).

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Le but principal de cette these est donc de donner une image complete d’un plasma poussiereuxdepuis sa creation jusqu’a son extinction. Cependant, les travaux effectues dans chaque themeont ete motives par des resultats anterieurs qui meritaient d’etre approfondis :

1. Le premier theme (Chapitre 4) est consacre a la croissance de poudres par pulverisationdans des decharges RF capacitives et a son influence sur les parametres du plasma. Cetravail est influence par les activites du groupe du Prof. Goree [59, 73, 74] et est dans lacontinuite des experiences menees par Dr. Mikikian au GREMI [61, 66, 68, 75]. Son prin-cipal but est la comprehension de l’influence de la croissance de poudres sur les parametresdu plasma et de la decharge. Pour cela, la formation de poussieres par pulverisation aete etudiee dans deux reacteurs differents : le reacteur PKE-Nefedov et le reacteur Com-PLExS. Des diagnostics electriques (tension d’autopolarisation et amplitude du courant),des diagnostiques optiques (imagerie video rapide, spectroscopie, luminosite du plasma)et des mesures de sondes de Langmuir ont ete utilises. Les resultats montrent que latension d’autopolarisation decroıt en valeur absolue durant la croissance des poudres etque l’amplitude du courant diminue. Cela est attribue a la perte des electrons libres duplasma a la surface des poussieres en formation. La fonction de distribution en energie deselectrons (FDEE) est profondement modifiee par la presence des poudres (accroissement dela temperature electronique mais perte des electrons tres energetiques). La purete du gaz dela decharge affecte enormement la croissance des poussieres. Finalement, des instabilitesdu plasma durant la croissance des poudres ont ete observees. Celles-ci suivent un schemad’evolution temporel bien precis.

2. Le deuxieme theme (Chapitre 5) est centre autour des instabilites d’un nuage de poussieresdans un plasma complexe et plus particulierement les instabilites du « void ». Nos resultatss’interessent plus particulierement aux battements (heartbeat) du « void »(dilatations etcontractions regulieres de la taille du « void »), qui ont ete etudie theoriquement [58,76] et experimentalement [60, 61], et aux instabilites dues a la croissance d’une nouvellegeneration de poudres a l’interieur du « void » : dilatations et contractions de la tailledu void de grande amplitude et basse frequence et qui se terminent par l’apparition d’unnouveau nuage de poussieres dans l’ancien « void », et le « void »en rotation (revolution du« void »autour de l’axe de symetrie du reacteur). Pour ces differentes etudes, nous avonsparticulierement mis l’accent sur la mesure de certains parametres de la decharge (courant)ainsi que sur des mesures spatialement resolues de la luminosite du plasma (imagerievideo rapide, fibres optiques focalisees et couplees a des photomultiplicateurs) et de ladynamique du nuage de poudres (imagerie video rapide). La correlation entre ces differentesmesures a permis une meilleure comprehension des phenomenes physiques impliques dansces instabilites. Les contractions du « void »sont accompagnes d’un fort accroissement de laluminosite au centre du plasma et donc de l’ionisation. De plus, des contractions avorteesdu « void »ont ete observees entre les sequences principales de contraction-expansion. Lacinetique d’evolution des dilatations et contractions de la taille du void de grande amplitudeet basse frequence est liee a la cinetique de croissance du nouveau nuage de poudres.

3. Le troisieme theme (Chapitre 6) est consacre aux etudes sur la phase post-decharge desplasmas complexes et sur la charge electrique residuelle des poudres quand le plasmaa completement disparu. Ce travail fut initie par la decouverte de la charge electriqueresiduelle des poussieres apres l’arret d’une decharge RF capacitive lors d’experiencesen microgravite [77]. Avant ce resultat, on pensait que les poudres perdaient toutes leurscharges pendant la phase post-decharge et devenaient electriquement neutres. Par conse-quent, des etudes complementaires etaient necessaires.Cela a naturellement donne lieu a des recherches sur la phase post-decharge des plasmascomplexes afin de comprendre les processus de dechargement des poudres. Comme la chargedes poudres est fortement liee a la diffusion des porteurs de charge dans la phase post-decharge, des etudes theoriques sur l’influence des poussieres sur la perte des ions et deselectrons ont ete menees. La comparaison avec les resultats experimentaux disponibles [21]montre que la mesure du temps de disparition du plasma durant la post-decharge peut etre

34


utilisee comme un diagnostic complementaire. Des mesures de la decroissance de la densiteelectronique dans differents plasmas avec ou sans poudres ont montre que ces dernieres ontun effet consequent sur la diffusion des electrons durant la phase post-decharge.Afin d’approfondir les connaissances sur la charge residuelle des poudres, des experiencesau sol ont ete realisees dans le reacteur PKE-Nefedov. Les poudres ont ete formees direc-tement dans le plasma et maintenues entre les deux electrodes, apres que la decharge a eteeteinte, par une force de thermophorese compensant la force de gravite et engendree parun systeme inspire des experiences de Dr. Rothermel [78]. Nous avons confirme que lespoudres portent bien une charge electrique residuelle apres l’arret de la decharge et avonsdecouvert que cette charge peut etre positive ou negative. Des fonctions de distributiondes charges residuelles ont ete reconstruites en mesurant les charges d’un grand nombrede poudres. Leur variance est plus grande que celle attendue pour la fonction de distri-bution de charges des poussieres dans un plasma allume. Les simulations du processus dedechargement ont montre que la fonction de distribution des charges residuelles est tressensible a la diffusion des ions et des electrons et plus precisement a la transition de ladiffusion ambipolaire vers la diffusion libre.

En conclusion, cette these s’interesse a l’influence des poudres sur des decharges RF de-puis leur allumage jusqu’a leur extinction. Une analyse detaillee de l’effet des poussieres auxdifferents instants de la vie du plasma est realisee. Les principaux resultats s’articulent autourdes differentes etapes de la vie de la decharge :

1. L’effet de la croissance des poudres a l’allumage de la decharge et son influence sur lesparametres du plasma ainsi que les instabilites associees.

2. Les instabilites du « void »dans un nuage de poudres bien etabli.

3. L’effet des poussieres sur la phase post-decharge et l’influence de la diffusion du plasmasur le dechargement des poudres et les charges residuelles.

35

Chapter 2

Review of complex plasma physics

Dusty or complex plasmas are partially ionized gases composed of neutral species, ions, electrons

and charged dust particles. Dust particles are electrically charged due to their interactions with

ions and electrons of the surrounding plasma [28, 79, 80]. In this chapter, the charging process

of the dust particles and the forces acting on them are described. New phenomena due to the

presence of dust particles in a plasma are also discussed.

2.1 Charge and charge distribution

The particle charge controls the dust dynamics both in laboratory and industrial plasma reactors,

and also in space plasmas. Thus one of the main challenges in dusty plasma is to understand the

charging of dust particles in a wide range of experimental conditions, which includes industrial

and space plasmas. For example, in most industrial processes in the microelectronics industry,

which uses silane as the reactive gas, dust contamination is a vital problem [5]. The dust dy-

namics and coagulation in space plasma are also phenomena governed by dust particle charges

[81, 82].

Elementary processes leading to dust particle charging are the collection of the ions and the elec-

trons of the surrounding plasma, the interactions with photons, secondary electron emission due

to collisions with energetic particles, thermionic emission, etc. In laboratory plasmas, charging

by ion and electron collection is the prevalent effect.

Two extreme cases can be considered. When the Debye length λD of the dusty plasma is much

smaller than the intergrain distance a, the dust particles can be considered as isolated in the

plasma. When the Debye length λD is larger than the intergrain distance a, the dust particles

interact with each other and have a collective behaviour. The size of the dust particle compared

to the Debye length also has an influence. Here, we consider only the case where rd ≪ λD where

rd is the radius of the dust particles.

36

2.1 CHAPTER 2. REVIEW OF COMPLEX PLASMA PHYSICS

2.1.1 The Debye length

A fundamental characteristics of plasmas is the screening of electric field created by individual

charged particles (ions, electrons and dust particles) or by a surface with a potential that is

different to the plasma potential. This screening effect has a characteristic length: the Debye

length. This is the length below which the electric field of a charged particles can influence the

other charged particle. In dusty plasmas, the linearised Debye length is [5, 83]:

λD =λDeλDi√λ2

De + λ2Di

(2.1)

where λDi(e) is the ion (electron) Debye length defined as:

λDi(e) =

√ǫ0kBTi(e)

ni(e)0e2(2.2)

with ǫ0 the vacuum permittivity, kB the Boltzmann constant, Ti(e) the ion (electron) tempera-

ture, ni(e)0 the ion (electron) density and e the elementary charge.

For typical plasmas with Te ∼ 3 eV , Ti ∼ 0.03 eV and ni0 ≃ ne0 ∼ 1015 m−3, the Debye length

is λD ∼ 40 µm

2.1.2 Isolated dust grains

Orbital motion limited (OML) theory for a spherical dust grain

The charging (decharging) process of dust particles in a plasma is governed by the contributions

of all currents entering (or leaving) the dust surface, involving plasma electron and ion currents,

photoemission and thermionic emission currents, etc.

dQd

dt=

∑

a

Ia +∑

l

Il (2.3)

where Qd is the charge of the dust particle and Ia and Il are currents absorbed and emitted by

the dust particle (with appropriate signs). For discharge plasmas we can ignore the emission

current in most cases. The plasma ions and electrons are collected by the grain, which acts like

a floating probe. Dust particles are consequently electrically charged by accumulation of plasma

particles “falling” onto their surfaces. Dust particle electric charge is determined by:

dQd

dt=

∑

j

Ij (2.4)

where Ij is the current associated with the plasma j−species.

Let’s consider a spherical dust grain with radius rd immersed in an ion and electron plasma. It

is obvious that when vTe ≫ vT i where vTe(i) =√

8kBTe(i)/πme(i) is the electron (ion) thermal

speed with me(i) the electron (ion) mass, the electrons reach the dust grain surface at a greater

37


b

v

v

r

j

j

gj

d

Figure 2.1: Schematic of interaction of a particle j with a dust grains of the same charge

rate. Consequently, more electrons are collected and dust grain surface potential φg is negative.

Ion and electron currents are of course affected by the dust surface potential. Thus, when it

is too repulsive for the j−species, the current from the oppositely-charged plasma particle is

favoured which modifies, in proportion, the dust surface potential.

When rd ≪ λDi ≪ λmfp where λmfp is either the electron or ion-neutral collisional mean

free path, the current of the j−species on the dust particle can be calculated using the orbital

motion limited (OML) theory [84, 85]. The two major assumptions of this model are [86]:

• Any plasma particle initially far from the dust particle can reach the surface of the dust

particle and stick to its surface if it is allowed by the conservation laws. Moreover, this

process is independent from the structure of the electrostatic potential near the particle.

• For a spherical dust particle, the limiting impact parameter of the plasma particle corre-

sponds to a trajectory which is tangential to the dust particle.

Let’s consider a plasma particle j approaching a dust particle from infinity (fig.2.1). When this

particle enters the Debye sphere of the dust grain, it experiences a force due to the electrostatic

field of the grain and is deflected. Taking vj as the speed of the particle j before interaction, vgj

its speed afterward and bj the impact parameter, it is obvious that for fixed vj , if bj reduces,

the particle j collides with the dust grain. The cross section for a j particle-dust grain collision

σdj is:

σdj = πb2

j (2.5)

As dust grains are massive compared to ions and electrons (typically md ≫ mi ≫ me), momen-

tum and energy conservation give:

mjbjvj = mjvgjrd (2.6)

1

2mjv

2j =

1

2mjv

2gj +

qjQd

4πǫ0rd(2.7)

where mj is the mass of the j particle and qj its charge. The dust charge Qd is directly

linked to the potential difference φd = φg − Vp where Vp is the plasma potential. One can

38


write Qd = C · φd where C is the capacitance of the dust grain. For a spherical dust grain,

C = 4πǫ0rd exp(−rd/λD) ≈ 4πǫ0rd with λD ≫ rd. With Eqs.2.5, 2.6 and 2.7, one can obtain:

σdj = πr2

d

(1 − 2qjφd

mjv2j

)(2.8)

If fj(vj) is the j−species velocity distribution function, the current Ij on dust grains is:

Ij = qj

∫ ∞

vjmin

vjσdj fj(vj)d

3vj (2.9)

Two cases must then be considered:

• if qjφd < 0, the interaction force between the j particle and the dust grain is attractive

and vjmin = 0

• if qjφd > 0, the force is repulsive and (1/2)mjv2jmin

= qjφd ⇒ vjmin =√

2qjφd/mj

For a Maxwellian plasma, the velocity distribution function is:

fj(vj) = nj

( mj

2πkBTj

)3/2exp

(−

mjv2j

2kBTj

)(2.10)

with nj the density of the plasma j−species. Thus, in the attractive case, one can obtain:

Ij = 4πr2dnjqj

( kBTj

2πmj

)1/2(1 − qjφd

kBTj

)(2.11)

and in the repulsive case, one can obtain:

Ij = 4πr2dnjqj

( kBTj

2πmj

)1/2exp

(− qjφd

kBTj

)(2.12)

When the dust particle charge reaches steady state, ion and electron currents, Ii and Ie respec-

tively, compensate each other:

Ie = Ii (2.13)

thus, for negatively charged dust particles (which is generally the case in most laboratory plasma

where Te > Ti), the dust potential upper limit is thus [5, 83, 86]:

φd =kBTe

eln

(ni

ne

(meTe

miTi

)1/2)

(2.14)

and the dust particle charge is:

Qd =4πǫ0rdkBTe

eln

(ni

ne

(meTe

miTi

)1/2)

(2.15)

39


Eq.2.15 gives the maximum charge on an isolated dust particle. Matsoukas and Russel [79]

showed that this expression must be corrected by a factor Cc depending on plasma parameters

giving:

Qd = Cc ·4πǫ0rdkBTe

eln

(ni

ne

(meTe

miTi

)1/2)

(2.16)

For typical argon plasmas (me/mi ≈ 1.4 · 10−5 and 1 < Te/Ti < 100, this factor is Cc ∼ 0.73.

For Te/Ti > 1000, Eq.2.16 is a very poor approximation and completely breaks down when

Te/Ti ! mi/me which are not cases of practical interest.

Influence of trapped ions

In the previous section, the effect of the collisions of the electrons and ions on the dust charging

process and the electrostatic shielding have been neglected. However, under some conditions,

a significant amount of trapped ions due to charge exchange collisions with the neutral gas

background can surround the dust particle. Hence the positive ion current can be significantly

increased and the shielding properties of the grain are changed. It has been shown that when

Ti ≪ Te (nearly all newly created ions are trapped in the potential well of the dust particle) and

when r2d/λ2

D ≪ Ti/Te (few newly created ions fall immediately on the dust particle), the density

of trapped ions can be large and the dust particle potential magnitude significantly reduced (as

much as 50 %) [87, 88].

Dust charge distribution

The formalism developed above gives the charge of a dust particle in a steady state plasma. It is

obvious that, when the dust particle has its equilibrium charge, the average number of collected

electrons is equal to the average number of collected ions but since the charging of a dust particle

is a discrete process, electrons and ions are captured at random times. The number of charges

on a dust particle Zd = |Qd/e| fluctuates around its equilibrium value given by the condition

that the total current to the particle is zero. Charge fluctuations and dust charge distributions

(DCDs) have been the subject of many theoretical studies ([79, 89, 90] and references therein).

In the most fundamental process, the charge fluctuation is of magnitude ±e, each time an

electron or an ion is captured by the dust particle. A Fokker-Planck description of the charging

process in weakly ionised gases and fluctuations arising from the nature of this process have

been presented by Matsoukas and Russel [91]. It gives time for charge fluctuations τQ, the mean

time for collisions of a dust particle with a charge species τc as well as the variance σ of the

mean charge Qmean on a dust particle.

We consider W = W (Qd, t) to be the probability for a dust particle to carry the charge Qd

at time t. If it is assumed that the charging currents depend on the instantaneous charge on

the dust particle and not on any prior history, the particle charge constitutes a Markov process

40


whose master equation can be deduced from Eq. 2.4:

∂W

∂t

Qd

= [IiW |Qd−e − IeW |Qd+e − (Ii − Ie)W |Qd]/e (2.17)

Assuming that the dust particle charge is a continuous variable and linearising the currents

in the vicinity of the steady-state charge Qmean which is defined by Eq.2.13, Eq.2.17 can be

rewritten as [91]:∂W

∂(t/τQ)=

∂(Qd − Qmean)W

∂Qd+ σ2 ∂2W

∂Q2d

(2.18)

where τQ and σ2 are defined as:

τQ = 1/(−Ie′ − Ii

′) (2.19)

σ2 =e

2

( Ii − Ie

−Ie′ − Ii

′

)(2.20)

The primes indicate the first derivative with respect to Qd and the bars indicate calculation at

Qd = Qmean. Eq.2.18 is mathematically identical to the Fokker-Planck equation for the diffusion

of a particle in a harmonic potential and views the charge as a stochastic variable diffusing in

the charge space and opposed by a force produced by the deterministic charging current. The

solution of this equation is (see Ref.[91] and App.A):

W (Qmean + x, t) =1

σ√

2π(1 − e−2t/τQ)exp

(− (x − x0e

−t/τQ)2

2σ2(1 − e−2t/τQ)

)(2.21)

where x = Qd − Qmean. From Eq.2.21, it is clear that the charge distribution is Gaussian at

all the times and approaches a steady state with a time constant τQ, with a mean dust particle

charge Qmean and a variance σ which depends on the charging currents. The variance σ can be

written as: (σ

e

)2=

τQ

2τc(2.22)

where τc = e/(Ii − Ie). It has been shown that τQ represents the time for charge fluctuations

and τc represents the mean time for collisions between a dust particles and a charge species [91].

In the case of Maxwellian current, the time for charge fluctuations τQ is [91, 92]:

τQ =( 4λ2

i

vT ird

) 1

1 + Ti/Te − eQ/4πǫ0rdkBTe(2.23)

Fokker-Planck simulations [93] were performed for 190 nm radius particles (experimental dust

radius obtained during dust particle residual charge measurements; see Chap.6) with Te = 3 eV,

Ti = 0.03 eV and n = ne = ni = 5 · 109 cm−3 (see Fig.2.2). It shows that the dust particle

charge evolves to an equilibrium charge in a time τQ ∼ 4 µs which is in agreement with Eq.2.23,

and that the charge distribution at equilibrium is a Gaussian. Moreover, the mean charge at

equilibrium Qmean is in agreement with the theoretical prediction of Eq.2.16 (which predicts an

41


!!""" !#$" !#%" !#&" !#'" !#"""

!"

'"

("

&"

)"

%"

*"

Charge (e)

Num

ber

ofdust

part

icle

s

Qd = −952e

σ(Qd) = 17eσ(Qd)√

|Qd|= 0.55

!

µ"#

! $ %! %$ &! &$ '!

()*+

,-./0+12)31+#

!%!!!

!4!!

!5!!

!6!!

!7!!

!$!!

!8!!

!'!!

!&!!

!%!!

2

Figure 2.2: Calculation of dust particle charges using a Fokker-Planck algorithm [93] for 190 nmradius particles in a plasma with Te = 3 eV , Ti = 0.03 eV and n = ne = ni = 5 · 109 cm−3.Left: Dust particle charge distribution for 500 iterations. Right: Dust particle charge evolutionwhen immersed in the plasma

equilibrium charge Qmean = −951.9e).

Similar results were obtained by Tsytovich et al [89] by assuming the discreteness of the elec-

tron and ion distributions. Khrapak et al [94] obtained the magnitude of fluctuation in terms of

the temporal autocorrelation function and reported dependence of the dust charge distribution

(DCD) on plasma parameters. All proposed models confirm that the DCD is Gaussian and that

the standard deviation of the stochastic charge fluctuation varies as σ(Qd) ≃ δ√|Qd|, where Qd

is the dust particle charge, Qd is the mean dust particle charge and δ is a parameter depending

on plasma conditions and is close to 0.5. Computer simulations of the charging process also

report the same result [56, 93, 95].

Nevertheless, no direct measurement of the DCD has been performed due to the shortness of the

charging and characteristic charge fluctuation time scales (in the µs range). So at any particular

moment an ensemble of monosized dust particles is characterized by their DCD. However, on

the dust time scale (in the ms range) the behaviour of a dust particle is determined by its mean

equilibrium charge. Existing experimental techniques for dust charge measurement based on the

analysis of dust dynamic response [96–98] can only determine the mean charge which is equal for

all monosized particles. However, charge fluctuations may have some impact in dusty afterglow

plasma and DCD may be measured in the late discharge afterglow (see Chap.6).

2.1.3 Cloud of dust particles

When the dust particle density in the plasma is high, the global charge carried by the dust

particles can be a significant fraction of the global charge carried by the ions and the electrons.

In this case, the neutrality condition has to be rewritten in order to take into account the dust

42


particles:

ni = ne + Zdnd (2.24)

where Zd = |Qd/e| is the dust particle charge number and nd the dust particle density. As can

be seen, it is possible that the electron density differs significantly from the ion density and it

must be taken into account in the charging equation. At steady state ion and electron currents

compensate each other. Consequently, combining Eqs.2.11, 2.12 and 2.24, the dust particle

charge can be obtained solving:

(Ti

Te

me

mi

)1/2(1 +

Te

Tiϕ) exp(ϕ) = 1 − PH (2.25)

where PH = Zdnd/ni is the Havnes parameter and ϕ is the dimensionless dust particle surface

potential. The latter can be linked do the dust charge number Zd by:

ϕ =Zde

2

4πǫ0rdkBTe(2.26)

Eqs.2.25 and 2.26 must be solved numerically.

2.2 Forces acting on dust particles

The dust particles are subjected to many forces when immersed in a plasma. In laboratory

discharges, the main forces are the gravitational force, the electric force, the neutral drag force,

the ion drag force and the thermophoretic force. In most laboratory cases, the radiation pressure

force can be neglected.

2.2.1 The electric force

As dust particles acquire an electric charge when immersed in a plasma, they are subject to an

electric force Fe induced by the electric field E that may exist in the plasma. In capacitively

coupled RF discharge, due to higher mobility of electrons, electric fields are greater in the sheath

region above the electrodes. A small electric field due to ambipolar diffusion of charged particles

toward the wall can also exist. Generally, in a plasma, the electric force Fe acting on dust

particles can be written as:

Fe = QdE (2.27)

2.2.2 The neutral drag force

This force results from collisions with neutral gas atoms/molecules. It is therefore proportional

to the gas pressure. It is the rate of momentum transfer between the neutral atoms/molecules

and the dust particles during their collisions. Assuming that the interactions between the dust

particles and the neutral gas atoms/molecules are hard sphere, elastic collisions, the neutral

43


drag force Fdn can be approximated by [99]:

Fdn ≃ nnmnσnd(vd − vn)2 (2.28)

where nn is the density of neutral atoms/molecules, mn their mass, σnd the cross section of the

dust neutral interaction, vd the average velocity of the dust particle and vn the average velocity

of the neutral atoms.

More accurate expressions of the neutral drag force have been developed. When the relative

speed |vd − vn| is very small in comparison with the neutral thermal speed vTn << 1 (Epstein

limit), the neutral drag force Fdn can be approximate as [83]:

Fdn = −8

3

√2πr2

dmnnnvTn(1 + αacπ

8)(vd − vn) (2.29)

where vTn =√

8kBT/πmn is the thermal speed with T the neutral gas temperature, and αac is

the accommodation coefficient. This coefficient would be zero for specular reflection and unity

for perfect diffuse reflection.

2.2.3 The ion drag force

The ion drag force describes how momentum is transferred from ions to the dust particles. This

force consists mainly of two components [42, 100, 101]: the force due to direct impact of ions,

i.e. the collection force Fcolldi and the force due to electrostatic Coulomb collision Fcoul

di :

Fdi = Fcolldi + Fcoul

di (2.30)

If vi >> vd where vi is the ion drift velocity, the relative dust-ion velocity can thus be approx-

imated by vi. In the usual derivation of ion drag force [42], it is supposed that there are no

interactions between the ions and the dust particle outside the Debye sphere. Moreover, it is

supposed that the ion mean free path is larger than the Debye radius (valid for low gas pressure).

The collection force is therefore:

F colldi = πnimivivsb

2c (2.31)

where

vs =(8kBTi

πmi+ v2

i

)1/2(2.32)

is the mean velocity of the ions approaching the dust particles and

bc = rd

(1 − 2qiφd

miv2s

)1/2(2.33)

is the maximum impact parameter given by the OML theory.

The Coulomb force is:

F couldi = 4πnimivivsb

2π/2Γc (2.34)

44


where

bπ/2 =eQd

4πǫ0miv2s

(2.35)

is the impact parameter for 90 deflection and

Γc =1

2ln

(λ2D + b2

π/2

b2c + b2

π/2

)(2.36)

is the Coulomb logarithm integrated over the interval [bc, λD].

It has been shown experimentally that this expression for the ion drag force is a good approxi-

mation [102]. However, better expressions for the ion drag force can be found in the literature

[103, 104]. In Ref.[103] , the fact that the ion-dust particle interactions can have a length range

larger than the Debye length is taken into account and shows that the ion drag force is increased

in comparison with the analytical results of Eqs.2.31 and 2.34. In Ref.[104], the effect of the

ion-neutral collisions, the distortion of the potential around the grain, the effects of the finite

size of the dust particles and the charging collisions are taken into account by combining the

binary collision approach with the linear kinetic formalism. Calculations for typical RF dis-

charge parameters have shown that the ion drag force has a complex dependance on the ion

flow velocity. The dependence of the ratio of the ion drag-to-electric force on the electric field

strength (or the ion flow velocity) decreases rapidly with the electric field. Thus the ion drag

force can only be important for subthermal and slightly suprathermal ion flows (when the flow

is due to the global electric field).

2.2.4 The gravitational force

In ground based experiments, dust particles are subjected to the gravitational force. This force

is proportional to the mass of the dust particle, hence to its mass density ρ and to its volume:

Fg = mdg =4

3πρr3

dg (2.37)

This force can be neglected for very small dust particles immersed in a plasma (rd < 1 µm) but

not for particles that are micrometer size or larger. Furthermore, this force can not be ignored

in afterglow plasmas (when the discharge is switched off) whatever the size of the particles since

the electrostatic and the ion drag force quickly disappear.

2.2.5 The thermophoretic force

A small dust particle suspended in a gas with a temperature gradient is subjected to a force

which will induce motion of the dust particle, in the absence of gas flow, from the hot region

to the cold region. This force called the thermophoretic force results from the difference in

momentum transfer to the dust particle by the gas molecules colliding with it from the cold and

hot side (see Fig.2.3). The expression for the thermophoretic force must be chosen carefully.

Indeed, it depends strongly on the Knudsen number Kn = lg/rd [105] where lg is the mean free

45


path of buffer gas species. In our experiment, we worked at an operating pressure around 1

mbar. In a previous paper [61], the size of grown dust particles was reported between 200 nm

and 800 nm. It gives, using results from Varney [106] for an atom-atom cross section, a Knudsen

number of 250 < Kn < 1000. Consequently we operate in the free molecular regime where a dust

particle is similar to a very large molecule. Many theories have been developed [105, 107–111]

and used [78, 105, 112] for thermophoresis in the free molecular regime. The most commonly

used equation is the Waldmann equation [107] which has been verified experimentally [113, 114]:

FT = −32

15r2d

ktr

vTn∇Tn (2.38)

where ∇Tn is the temperature gradient in the gas, and ktr is the translational part of the thermal

conductivity given by, for a monoatomic gas [115]:

ktr =15kB

4mnµref

(T

Tref

)ν

(2.39)

where µref is the reference viscosity at reference temperature Tref = 273 K and the exponent

ν results from a best fit of experimental viscosity near the reference temperature. For argon,

µref = 2.117 · 10−5 Pa·s and ν = 0.81 [115].

COLD SIDE

!

T 8

molecule

velocity

TF

HOT SIDE

Figure 2.3: A dust particle immersed in a gas with a temperature gradient

2.2.6 Order of magnitude of the different forces

The strength of the different forces acting on the dust particles depends on different power laws

of the dust particle radius. The electric force Fe is proportional to the radius rd (because the

charge Qd is proportional to the dust radius (Eq.2.16)). The ion drag force, the neutral drag

force and the thermophoretic force are proportional to r2d. The gravitational force is proportional

to r3d. Consequently it is clear that for small particles, the electric force will be dominant while

46


for big particles the gravitational force will become more important.

In Ref.[5], the orders of magnitude of the different forces in a RF argon plasma, with n = 5 · 109

cm−3, Tn = Ti = 500 K, Te = 3 eV, the local electric field at the sheath boundary is 30 V/cm

and the gas temperature gradient is 10K/cm, have been calculated for 100 nm and 1 µm dust

particles with a density ρ =2g/cm−3. The results are summarised in Tab.2.1.

Table 2.1: Order of magnitude of different forces acting on dust particles for different dustradius (from Ref.[5]).

rd = 100 nm rd = 1 µm

Electric force ∼ 2 · 10−13 N ∼ 2 · 10−12 NNeutral drag force ∼ 10−15 N ∼ 10−13 N

Ion drag force ∼ 5 · 10−14 N ∼ 10−12 NThermophoretic force ∼ 10−15 N ∼ 10−13 NGravitational force ∼ 10−16 N ∼ 10−13 N

2.3 Instabilities and waves in complex plasmas

2.3.1 Characteristic frequencies of complex plasmas

An important property of a plasma is its natural oscillations. When the plasma deviates from

equilibrium, i.e. the neutrality condition ne+Zdnd = ni is not satisfied (from a macroscopic point

of view), an electric field is created due to the space charge and tends to reestablish equilibrium.

Due to their different masses, charged species in the plasma do not respond equivalently.

Electron frequency

The electrons are accelerated by the electric field in order to reestablish quasi neutrality. Due

to their inertia, they pass the equilibrium point and a new electric field is created in the oppo-

site direction accelerating electrons the other way. Electrons oscillate around their equilibrium

position with electron angular frequency ωpe:

ω2pe =

nee2

meǫ0(2.40)

For an argon plasma with ne ≃ 1010 cm−3, the angular frequency is ωpe ≃ 6 GHz.

Ion frequency

Ions are heavier than electrons and so their response to an electric field is much slower. The ion

angular frequency ωpi is:

ω2pi =

niZ2i e2

miǫ0(2.41)

47


For an argon plasma with ni ≃ 1010 cm−3, the angular frequency is ωpi ≃ 20 MHz.

Dust frequency

As a charged species, the dust particles also respond to electric field. The plasma dust angular

frequency ωpd is:

ω2pd =

ndZ2de2

mdǫ0(2.42)

For dust particles with mass md ≃ 10−16 kg, Zd ≃ 1000 and nd ≃ 2 · 105 cm−3, the plasma dust

angular frequency is ωpd ≃ 2 kHz.

2.3.2 Examples of waves and instabilities observed in complex plasmas

In plasmas, a wide variety of waves exists due to coherent motion of the different plasma species

(Langmuir waves and ion acoustic waves for example). In complex plasmas, the charged dust

grains change the wave propagation due to inhomogeneities in the dust distribution, the modified

quasi-neutrality condition (Eq.2.24) and some considerations about dust particle dynamics. Two

major subclasses of waves exist in a complex plasma:

1. Waves and oscillations arising from the collective motion of the plasma particles (ions or

electrons) and affected by the dust particles

2. Waves and oscillations due to the collective motion of the dust particles in various dust

structures (dust cloud, liquids or crystals)

Dust Acoustic (DA) waves

The DA waves come from the collective motion of the dust grains. The DA wave velocity

is much smaller than the ion and electron thermal speeds and consequently ion and electron

inertia can be neglected and the DA wave potential can be considered to be in equilibrium. The

pressure gradient can be assumed as balanced by the electric force and thus the ion and electron

distributions are Boltzmann distributions. The restoration force in the DA waves comes from

the ion and electron (considered without inertia) pressure while the dust particle mass provides

the wave inertial support.

DA waves have been observed in the laboratory [52, 116]. The reported frequencies are around

10-20 Hz.

Dust-ion acoustic (DIA) waves

The DIA waves come from the collective motion of the ions affected by the presence of the dust

particles. For negatively charged dust particles (ni0 > ne0), the DIA wave phase velocity ω/k

is greater than the ion acoustic speed. It can be explained by an increase of the electron Debye

length due to electron losses on dust particles. Consequently, the electric fileld E = −∇Φ is

48


larger. As kvT i ≪ ω ≪ kvTe, Landau damping of DIA waves is negligible. DIA waves have been

observed in the laboratory and the typical frequencies are tens of kilohertz [117, 118].

Other waves and instabilities

Other varieties of waves can be found in complex plasma (from individual oscillations of dust

particles in sheaths of gas discharge to compressional waves in dust crystals) and their properties

(instabilities, damping, etc.) depend on the plasma parameters (see Ref.[62] and references

therein). Instabilities can also be self excited in complex plasmas. Thus, it has been reported that

the growth of dust particles in a plasma can trigger some instabilities [59, 66, 119] . Instabilities

of the void region (centimetre-sized region completely free of dust observed in the centre of dusty

discharges) have also been reported [60]. The frequencies of these instabilities are generally in

the range 0-1 kHz and can thus be associated with the dust particles.

2.4 Conclusion

In this chapter, the basis of the physics of complex plasmas has been reviewed. The charging

processes of dust particles in a running discharge have been described as well as the main forces

acting on the dust particles. New phenomena that may occur in a plasma due to the presence

of the dust particles have been briefly discussed. This background will be used in the analysis of

the results of the following chapters. In chapter 4, the effect of dust particle growth and related

instabilities on the discharge and plasma parameters are related to the charging of the dust

particles and the dust particle kinetics. In chapter 5, void instabilities are directly related to

the dust particle kinetics. In chapter 6, residual dust charge and residual charge distribution in

the afterglow plasma are studied in details and hence the dust charging process is at the centre

of this research.

49


2.5 Résumé du !apitre en français

Dans ce chapitre, les bases de la physique des plasmas poussiereux sont rappelees :– Les plasmas poussiereux sont des gaz partiellement ionises contenant des poudres chargees.

Ces dernieres se chargent aux travers des interactions avec les ions et les electrons duplasma environnant. En laboratoire, les courants directs d’ions et d’electrons a la surfacedes poudres sont le principal mecanisme de chargement. A l’equilibre, les courants d’ionset d’electrons se compensent exactement. Dans les plasmas de laboratoire, la temperatureelectronique etant generalement plus elevee que la temperature des ions, les poudres sontchargees negativement. Suivant les conditions, les collisions des ions et des electrons avecles atomes neutres peuvent modifier le potentiel electrostatique autour des poudres. Enparticulier, les collisions ion-atome avec echange de charge peuvent aboutir au piegaged’ions autour des poussieres et accroıtre le courant positif d’ions et ainsi reduire la chargedes poudres. Enfin, le processus de chargement des poudres est un processus stochastiquese faisant par pas de ±e. Par consequent, la charge des poudres varie constamment autourde la valeur d’equilibre et la distribution de charge est une gaussienne centree autour dela charge moyenne dont la variance depend des courants incidents d’ions et d’electrons.

– Les poussieres sont soumises a differentes forces dans les plasmas complexes. Les princi-pales sont la force electrostatique, la force de friction avec les atomes neutres, la force defriction avec les ions, la gravite et la force de thermophorese.

– Les poudres etant une nouvelle composante chargee du plasma, elles repondent a des per-turbations dont la frequence est bien plus faible que celles des electrons ou des ions en rai-son d’une masse tres importante. De plus, de nouvelles ondes et instabilites peuvent etrecreees grace a la presence des poussieres. Ces ondes se divisent en deux sous-categories :les mouvements collectifs des electrons ou des ions modifies par la presence de poudreschargees, et les mouvements collectifs des poussieres.

50

Chapter 3

Experimental Setups

3.1 Plasma chambers

3.1.1 The PKE-Nefedov Reactor

The PKE-Nefedov experiment was one of the first scientific experiments on board the ISS. The

chamber has been originally designed to study complex plasmas and plasma crystals under

microgravity conditions.

Plasma Chamber and pumping system

The PKE-Nefedov reactor is a capacitively-coupled RF reactor operating in push-pull mode. It

is a 10 × 10 × 5 cm3 glass vacuum chamber (Fig.3.1). The RF electrodes are stainless steel flat

circular plates with 4.2 cm diameter. A grounded dust dispenser is integrated in the centre of

the top electrode. The interelectrode spacing is 3 cm.

The RF source is a Dressler MPI-1 SPACE RF generator operating at 13.56 MHz which is driven

by two DC power supplies: a DC power supply (ISO-TECH Dual Tracking Model IPS230300)

delivering a 28 V DC voltage and 0.5 A DC current and, a DC Power Supply (AGILENT

E3610A) delivering DC voltage in the range 0 − 2.40 V. The output RF power can be varied

from 0 W to 4 W . The RF generator is linked to the electrodes through an integrated matching

box. It has been especially developed for low RF powers, which are required to create stable

and large complex plasma structures and plasma crystals.

The pumping system consists of a turbomolecular vacuum pump (PFEIFFER, 80 L.s−1) asso-

ciated with a mechanical vacuum pump (ALCATEL, 8 m3.h−1). The base pressure is about

2 ·10−6 mbar. The pressure is measured using two gauges: a full range gauge (Pfeiffer PKR 251)

from 10−9 mbar up to atmospheric pressure and a capacitive gauge (MKS Baratron with

PR 4000S control unit) in the range 10−4 − 2 mbar.

The gas is injected into the chamber through a port at the top. When the desired pressure is

reached, the valve controlling the gas flow is closed. The experiments are performed without

gas flow.

51

3.1 CHAPTER 3. EXPERIMENTAL SETUPS

Argon inlet

ELECTRICAL DIAGNOSTICS

ELECTRICAL DIAGNOSTICS

To pressure gauge

To pumping system

Valve

Valve

DiodeLaser

RF Power

RF Power

LASER SHEETPLASMA CHAMBER

Plasma & Particles

glass

glass

dust dispenser

electrodesRFinsulator

10 cm

ground

ground

3 cm 5.40 cm

4.20 cm

Figure 3.1: Schematic of the PKE-Nefedov reactor

52


The dust dispenser

In the PKE-Nefedov chamber, the dust particles can be either grown or injected directly in

the plasma. Dust particle injection is achieved by means of a dust dispenser in the middle of

the upper electrode. It consists of a kind of salt shaker filled with dust particles and stainless

steel marbles. A grid closes the shaker. Dust particles are injected into the discharge region

between the electrodes by an up and down oscillating motion of the dispenser. The particles

are accelerated inside the reservoir and released through a sieve with a mesh size adapted to

the particle diameter. Four sizes of dust particles can be injected in the plasma, all made of

melamine formaldehyde: 3.4 µm, 1.02 µm, 500 nm and 200 nm diameter dust particles. A 10 µm

step sieve was used and, several grid layers were used for the smallest dust particles.

3.1.2 The methane and silane reactor

The PKE-Nefedov reactor was developed originally to perform microgravity experiments and

thus its design is very compact. Hence, it is very difficult to add in situ diagnostics. For the

studies of dusty afterglow plasmas (see Chap.6), it was necessary to measure directly the electron

density decay during the post-discharge phase. Spectroscopic measurement are not sufficiently

sensitive during the post-discharge due to the fast decrease of the signal. Consequently, only

diagnostics such as the microwave resonant cavity could be used to measure the decrease of

electron density during the post-discharge phase with good time resolution. However, this

diagnostic could not be adapted to the PKE-Nefedov reactor and thus two other identical RF

reactors were used, where dust particles are grown using reactive gases such as methane (CH4,

carbon based particles) or silane (SiH4, silicon based particles) which has been intensively used

to study dust growth mechanism and kinetics[34–36, 119, 120]. These reactors allowed us to

use in-situ diagnostics such as the microwave resonant cavity, and furthermore, as the growth

mechanisms are well-known (especially for argon-silane plasma), the size and density of the dust

particles could be chosen.

Experimental apparatus

The reactor is a capacitively coupled RF discharge box inside a vacuum vessel (Fig.3.2). The

discharge box is a grounded stainless steel hollow cylinder. Its inner diameter is 13 cm. A 20%

transparency grid with 1 mm diameter holes closes the base and allows a laminar and uniform

gas flow. On the lateral surface, four 2× 4 mm2 vertical slits give optical accesses to the plasma

in perpendicular directions.

The RF electrode is stainless-steel and shower head style. It provides the gas supply and

constitutes the upper surface of the discharge box. The latter is a cylinder 12.8 cm in diameter

and 1 cm in height closed with a grid which is similar to that of the anode. A stainless steel tube

provides the RF bias and gas supply. A diffuser composed of 3 parallel grids is placed inside the

RF electrode in order to provide a uniform gas supply. A ceramic sleeve surrounding the tube,

isolates the RF electrode from the grounded discharge box. The interelectrode space is 3.3 cm.

53


R.F.

plasma

pumping port

gas inlet

porthole

boxdischarge

vaccumchamber

Figure 3.2: Schematic of the Methane/Nitrogen(Argon) reactor

Two pressure gauges are connected to the reactor:

• A Baratron type capacitive gauge measuring the absolute pressure in the range 10−4-

2 mbar independently of the gas nature. This gauge is placed near the pumping port of

the chamber and allows us to monitor the operating pressure during the experiments.

• A Penning gauge measuring the base pressure down to 10−6 mbar.

The pumping system consists of two complementary subsystems. A chemical mechanical vacuum

pump (ALCATEL 2033 CP PLUS, 35 m3·h−1) maintains the operating pressure during the

experiments (in the range 0.1-1.5 mbar) and does the primary pump down to 10−3 mbar. When

silane is used, a continuous flux of nitrogen (N2) is injected in the pump in order to dilute the

SiH4 to less than 1% and avoid inflammation or explosion due to contact with ambient air.

This pump can be isolated with an electric floodgate and a manual floodgate. The base pressure

is achieved by a turbomolecular pump (PFEIFFER Vacuum TMH 260, 210 L/s) coupled to

a mechanical vacuum pump (ALCATEL, 20 m3/h) which can be isolated from the vacuum

chamber with floodgates.

The gas is supplied by two independent lines. One for the methane (or silane), the other for

the dilution gas (Ar, N2). The gas mixing is made at the entrance of the reactor. Fluxes

are controlled by mass flow controllers and measured in sccm (standard cubic centimeter per

minute). For our experimental conditions, the gas flow did not exceed 10 sccm for methane or

silane and 45 sccm for nitrogen or argon.

The RF excitation (13.56 MHz) is provided by a RF generator (GERAL ARF 101) with an

output power varying in the range 0-100 W. This generator is controlled by a PC acquisition

board furnishing a digital signal (TTL, Voltage 0 − 5 V) and can hence be pulsed. A matching

54


box ensures a good coupling of the RF power between the generator and the discharge box

minimizing the reflected power and optimizing the power coupled to the plasma.

3.1.3 The ComPLExS (Complex PLasma Experiment in Sydney) reactor

Plasma Chamber and pumping system

The ComPLExS reactor is a capacitively coupled RF discharge consisting of two parallel elec-

trodes enclosed in a grounded stainless steel cylindrical vacuum vessel (30 cm diameter and

30 cm high; see Fig.3.3). The bottom electrode is powered through a matching network and

the top electrode is grounded. The top electrode sits on three ceramic stands which are located

at the edge of the bottom electrode. The bottom electrode is 10 cm in diameter and the top

electrode is 11.5 cm in diameter. The gap between the two electrodes is 2 cm. The electrode

system stands in the centre of the vacuum vessel.

The pumping system consists of two parts. The first part consists of a mechanical vacuum pump

10 cm

Vdc probe Irf probe

Vrf probe

2 cm

11.5 cm

Matching Box To turbomolecular pump

Glass tube

Argon inlet

Ceramics

To rotary pump

RF power

Porthole

Insulator

Window

30 cm

30 cm

Figure 3.3: ComPLExS reactor schematic

(PFEIFFER DUO 016, 16 m3/h) directly linked to the vacuum vessel. The pumping speed can

be regulated by means of a remotely controlled electronic vacuum valve (MKS Control Valve

001-1575 and associated MKS 600 pressure controller). It maintains the operating pressure

during experiments and ensures the preliminary pump down of the chamber to 10−3 Torr. The

base pressure (∼ 5 · 10−6 Torr ) is achieved by a turbomolecular pump (BALZERS TPU-510,

510 L/s) coupled with the mechanical vacuum pump. This system can be isolated from the

55


vacuum chamber with floodgates.

The gas supply is ensured by an independent line. The gas flux is controlled by a mass flow

controller and measured in sccm. In our experimental conditions, the argon flow does not exceed

20 sccm. The pressure is monitored by two pressure gauge: a full range gauge (PFEIFFER Full

range gauge) and a MKS Barathron gauge (range 10−4 − 2 Torr).

The RF excitation (13.56 MHz) is provided by a RF generator (LEADER Standard signal gener-

ator 3215) coupled to an RF amplifier (EIN Model 3100L RF Power amplifier). A matching box

ensured good coupling of the RF power between the generator and the discharge minimising the

reflected power and optimising the power coupled to the plasma. The RF peak to peak voltage

on the powered electrode can be up to Vpp=350 V.

3.2 Diagnostics

3.2.1 Electrical diagnostics

Self-bias voltage

When the powered electrode (the cathode) in a discharge is connected to an RF generator

through a blocking capacitor (as is the case for capacitively coupled RF discharges), there

is some irreversible escape of negative electric charges into the electrode, and hence the gap

gas becomes positively charged [121]. In a symmetric discharge, the two electrodes operate

in identical regimes and so collect the same amount of charges. Consequently, they have equal

mean potentials with or without a blocking capacitor. However, for an asymmetric configuration

the amount of charges collected by the electrodes is different and if the circuit does not allow

direct current as is the case when there is a blocking capacitor, a different amount of charges is

accumulated by each electrode and a difference of potential results between the electrodes: the

self bias voltage. In a low pressure discharge, the asymmetry occurs naturally because there is

always more grounded surfaces than powered surfaces. In the usual case, the self-bias voltage

of the powered electrode is negative with respect to the ground and depends on a power law of

the electrode area ratio [20, 122].

The cathode voltage is thus the sum of the RF voltage VRF and the self-bias voltage Vdc:

VC(t) = VRF (t) + Vdc (3.1)

The self bias voltage is explained as follows. During the part of the RF cycle when the cathode

is positive compared to the plasma, there is an electron flux to the electrode and the capacitor

is charged to a negative potential. During the other part of the RF cycle, there is an ion flux

that tends to charge the capacitor to a positive potential. Because the ions are less mobile than

the electrons (me ≪ mi), the electron flux will be greater than the ion flux until the average

potential of the powered electrode reaches a negative value for which the ion flux balances exactly

the electron flux. Then, for most of the RF cycle the powered electrode is negative with respect

to the plasma and there is an ion flux to the electrode. For a small fraction of the cycle the

56


electrode is positive with respect to the plasma allowing sufficient electrons to flow so that there

is no net charge flux to the electrode over the RF cycle.

It has been observed that the self-bias voltage changes as a result of the effect of particle growth

in the discharge, and that this variation correlates with particle growth [119]. Hence, this

diagnostic has been used on the three different reactors. On the PKE-Nefedov reactor and the

Methane reactor, the self bias voltage signal was digitised with a numerical oscilloscope with a

minimal sampling rate of 5 kS/s. Due to device limitations, the sampling rate on the ComPLExS

reactor was around 25 S/s.

Amplitude of the fundamental harmonic of the RF current

The current in a RF capacitive discharge is the sum of the conduction current and the displace-

ment current. When there is no plasma, there is only the displacement current which corresponds

to the reactive current through the vacuum capacitor formed by the parallel electrodes (In reac-

tors in which the RF electrode is surrounded by a counter-electrode, the displacement current is

mainly due to the stray capacitance between the cathode and the counter-electrode because the

capacitance between the two electrodes is negligible due to the large distance between them).

When the discharge is turned on, the conduction current is to a first approximation proportional

to the plasma density n0 [121]. Consequently the evolution of the amplitude of the RF current

can give a good idea of the evolution of the plasma density during the life of the discharge.

Moreover, in asymmetric discharges, for a sinusoidal voltage applied across the electrodes, the

current waveform exhibits anharmonicity due to the different displacement current densities at

the electrodes caused by the asymmetry and the non-uniformity of the sheath (the ion density

decreases from the plasma towards the electrodes) [121]. Consequently, the current waveform

exhibits harmonics which depend on the discharge conditions. It has been shown that the evo-

lution of the 2nd or 3rd harmonics of the RF current is a good diagnostic to monitor the growth

of dust particles in a RF capacitive discharge [119, 123]. In the PKE-Nefedov, previous studies

have shown that the fundamental harmonic of the RF current is also sensitive to the dust par-

ticle growth [61]. Consequently, the amplitude of the fundamental harmonic of the RF current

has been monitored during experiments on the PKE-Nefedov reactor. The signal was digitised

with a numerical oscilloscope with a minimal sampling rate of 5 kS/s.

3.2.2 Optical and video diagnostics

These diagnostics have been used on the PKE-Nefedov reactor.

Optical fibres

Five optical fibres were focussed on a defined region of the plasma chamber using cylindrical

lenses, each collecting the light emitted by the plasma (integrated over all frequencies) in a

precise region of the discharge with a spatial resolution of ∼ 3 mm..

The fibres were coupled to a photmultiplier tube (PMT) array which allowed us to resolve fast

57


RF Generator

TV MonitorsCameras

Lasersheet

Laser diode

Optical fibers

(13.56 MHz)

Camera (Classic CCD or Fast Camera)

To pumping

Electrode

Electrode

RF Power

(Push−Pull)

system

(x1)

Argon injection

To photomultiplier tube array

(x1/3)

Particles

&

Plasma

Figure 3.4: Principle schematic of optical diagnostics

variation of the collected light intensity (one photomultiplier per fibre). The signal from each

PMT of the array was digitised with a digital oscilloscope with a minimal sampling rate of 5

kS/s.

CCD Cameras

A thin laser sheet perpendicular to the electrodes illuminated the dust particles. It was created

by a 685 nm wavelength laser diode and a cylindrical lens. This laser sheet could be widened

by adding a mirror on the other side of the reactor.

Two CCD cameras recorded the light scattered by the dust particles at right angles to the laser

sheet. Interference filter were placed in front of the camera lenses in order to reduce the plasma

glow light. Video signals were transferred to a computer via a frame-grabber card with 8 bit

grey scale and 560 × 700 pixel resolution. The first camera had a field of view over 8.53 × 5.50

mm2 confined to the centre of the chamber (approximately a third of the chamber) while the

other one had a field of view covering a larger part of the interelectrode space.

A third camera (SONY DXC-1850P Colour) recorded the scattered light at a small angle (∼ 30)

from the laser sheet. It allowed observations of the dust particles at an early stage of growth

because scattering is more efficient in the forward direction for small particles. An interference

filter was placed in front of the camera lens in order to reduce the light emitted by the plasma.

58


Fast Camera

Fast CCD cameras (Mikrotron MC 1310 and SciTech SpeedCAM MiniVis ECO-1) were used to

record the plasma glow light or the scattered laser light. In order to record the laser scattered

light, the camera was placed at the position of the SONY camera. An interference filter was

placed in front of the lens when only the dust cloud motion was recorded. The plasma glow

light could also be recorded from this position by removing the filter. When the laser sheet was

turned on, it was possible to resolve the plasma glow and the dust particle cloud at the same

time.

3.2.3 Laser extinction measurement

LENS

LASER

PM

Chamber wall

Electrodes

MIRRORS

To oscilloscope

ITERFERENCE FILTER

OPTICAL FIBRE

Figure 3.5: Laser extinction measurement schematic

When a laser beam travels through a dust cloud, the incident light intensity is attenuated.

The corresponding extinction parameter Cext is defined by the relation:

d

dxIt(x) = −Cextnd(x)I(x) (3.2)

where It(x) is the intensity of the incoming laser beam and nd(x) is the local concentration of

dust particles. The extinction parameter is linked to absorption and scattering phenomena:

Cext = Cabs + Csca (3.3)

The intensity at a given point x in the particle cloud can be obtained by integrating Eq.3.2:

It(x) = It(x0) exp[−

∫ x

x0

nd(x)Cext(x)dx]

(3.4)

where x0 is the entrance edge of the dust particle cloud. If the cloud is homogeneous in term of

extinction coefficient, the intensity of the light passing through the dust particle cloud is related

59


to the number of dust particles Nd along the light path:

It(l) = It0 exp(−NdCext) (3.5)

where l is the path length through the dust cloud. If the dust particle density is homogeneous,

the attenuation will give useful information about the dust particle density along the light path

as long as the extinction coefficient is known.

The extinction parameter is a function of the properties of dust particles (size, material, etc.).

For spherical particles, this coefficient is independent of the light polarisation. When the radius

of the dust particles is much smaller than the radiation wavelength (rd ≪ λrad), the absorption

and scattering coefficient are within the Rayleigh approximation:

Csca =128π5

3λ4rad

∣∣∣m2 − 1

m2 + 2

∣∣∣r6d (3.6)

Cabs =8π2

λradIm

(m2 − 1

m2 + 2

)r3d (3.7)

where m is the complex refractive index which depends on the properties of the dust particles

(material, structure, etc.).

Laser light extinction measurements have been performed in the PKE-Nefedov reactor with an

HeNe laser (λrad = 632.8 nm). The laser light passed through the dust particles cloud four times

to improve the extinction signal. A schematic of the experiment is presented in Fig.3.5.

3.2.4 Emission spectroscopy

Optical spectroscopy of plasma emission has been used in many experiments to deduce important

parameters of the plasma such as the electron temperature and the electron density (see for

example [124, 125]). The main advantage of plasma emission spectroscopy is that it is a non-

intrusive method. However, in order to extract the electron temperature and/or density in the

plasma, the kinetic processes that populate and depopulate the excited states of the species in

the plasma need to be carefully analysed.

In order to monitor the evolution of the electron temperature in the RF argon plasma of the

ComPLExS reactor, the intensity of several argon lines have been measured. Then using a

simplified steady-state collisional radiative model, referred to as corona equilibrium, the electron

temperature is deduced.

In the following section, the experimental setup will be described. Then, the corona equilibrium

model is described and the results that can be obtained from it are discussed.

Experimental setup

The simultaneous measurement of the intensity of argon lines while the discharge is running

has been done using two monochromators coupled to photomultiplier tubes (Newport Oriel

77348 and PR-1400RF 9902B/1034/1034; see Fig.3.6). The light coming from the centre of the

60


Oscilloscope

To pumping system

Porthole

Argon inlet

Lens

Lens

Electrodes

ceramics stands

PMT

PortholePorthole

Electronic vacuum valve

Monochromator

Monochromator

PMT

Figure 3.6: Emission spectroscopy measurement schematic

discharge was focused on the monochromator using converging lenses. Slits of defined width were

used in front of monochromator’s aperture. The intensities of three argon lines were measured:

750.38 nm, 415.86 nm and 518.77 nm. The slit width was 300 µm for the first line and 130 µm

for the two other lines1. The optical systems have been calibrated using a tungsten ribbon lamp

in order to be able to compare the relative intensities of the different lines.

Estimation of Te: corona equilibrium

If the electron density is sufficiently low, the balance between excitation from the ground state

by electron collisions and radiative de-excitation determines the population of excited states of

atoms and ions of the plasma. Using this simplified model (steady-state collisional radiative

referred to as corona equilibrium2.), the density of excited state nk is given by [124]:

nen1k1k = nk

∑

j>k

Akj (3.8)

where n1 is the ground state density, k1k is the electron impact excitation rate coefficient from the

ground state (level 1) to level k, and Akj is the transition probability for the radiative transition

1The slit was wider for the measurement of the intensity of the 750.38 nm line due to the lower sensitivity ofthe PMTs at this wavelength.

2For levels of interest corona equilibrium requires ne << 1011 cm−3 [126]. In our experiments, ne ∼ 108− 109

cm−3

61


k → j where j < k. The power radiated per unit of volume for the radiative transition from

level k to a lower level j is:

Ikj = hνkjAkjnk (3.9)

where hνkj is the energy gap between levels k and j with h the Planck constant and νkj the

frequency of the radiation. The substitution of nk using Eq.3.8 gives:

Ikj = hνkjAkjnen1k1k∑

j>k Akj(3.10)

An analytic approximation commonly used for k1k (when the free electron EEDF is assumed

Maxwellian) is [124, 127]:

k1k = 8.69 × 10−8 × α1k × Z−3efff1k × u

3/2a

u1k× ψa(u1k, β1k) cm3s−1 (3.11)

where ua = 13.6kBTe, u1k = (E1 −Ek)/kBTe with E1 the energy of the ground level and Ek the

energy of the k level, α1k is a constant with a value approximately equal to 1, Zeff = Z−N +1,

where Z is the atomic number and N is the number of bounded electrons, is the effective atomic

number and f1k is the absorption oscillator strength of the transition from the ground state 1

to excited state k. The function ψa(u1k, β1k) is given by:

ψa(u1k, β1k) =exp(−u1k)

1 + u1k×

[1

20 + u1k+ ln

(1.25 ×

(1 +

1

u1k

))](3.12)

where β1k∼= 1.

Grouping together all the constants (as we are only interested in dependence on Te):

k1k ∝ u3/2a

u1k× exp(−u1k)

1 + u1k×

[1

20 + u1k+ ln

(1.25 ×

(1 +

1

u1k

))](3.13)

This function will differ for each excited state owing to the dependence via u1k on the excited

state energy Ek. Using Eqs.3.10, it is possible to deduce the electron temperature from argon

line ratio Ik2j2/Ik1j1 . Indeed, the ratio depends only on the electron temperature Te:

Ik2j2

Ik1j1

=λk1j1Ak2j2

∑k1>j1

Ak1j1

λk2j2Ak1j1

∑k2>j2

Ak2j2

· k1k2(Te)

k1k1(Te)(3.14)

where λij is the wavelength of the radiative transition. The spectroscopic data for the selected

argon lines used in this thesis are presented in Tab.3.1. The dependence of the different line

ratios3 and the intensity of the selected lines compared to the intensity at a reference electron

temperature Teref= 3 eV and constant electron density ne are presented in Fig.3.7. As can be

3It should be noted that the computed value of the ratio is not the absolute ratio as we do not take intoaccount the value of some constants such as the absorption oscillator strength in our calculation of electronimpact excitation rate coefficient as we only want the dependence on Te.

62

3.2

CH

AP

TE

R3.

EX

PE

RIM

EN

TA

LSE

TU

PS

λkj Ej Ek configurations Akj∑

k>j Akj gk Number of

(nm) (eV) (eV) (108s−1) (108s−1) radiative transitions

415.86 11.54 14.52 3s23p5(2P 3/2)4s − 3s23p5(2P

3/2)5p 0.014 0.017 5 3

518.77 12.91 15.30 3s23p5(2P 3/2)4p − 3s23p5(2P

1/2)5d 0.014 0.035 5 6

750.38 11.83 13.48 3s23p5(2P 1/2)4s − 3s23p5(2P

1/2)4p 0.445 0.448 1 2

Tab

le3.1

:Spectrosco

pic

data

and

num

ber

ofrad

iative

transition

sfrom

the

kth

elev

elcorre-

spon

din

gto

the

selectedA

rlin

esem

ittedby

the

plasm

a

63


10−1

100

100

10−2

10−4

Te (eV)

a.u

.

I415.86

/I750.39

I518.77

/I750.39

(a)

1 2 3 4 5 6

20

40

60

80

100

Te (eV)

I/I T

e=

3eV

λ=750.39 nm

λ=518.77 nm

λ=415.86 nm

(b)

Figure 3.7: a) Line intensity ratio as a function of Te. b) Evolution of line relative intensity asa function of Te

seen, when the electron temperature increases, the considered line ratios are increasing. Also

note that the intensity of each line increases when the electron temperature increases.

3.2.5 Electron density measurement by microwave resonant cavity

This diagnostic was used on the methane and silane reactors.

Theory

The propagation of an electromagnetic (EM) wave through a dielectric medium is governed by

the dielectric constant of this medium which is related to the density of free charges interacting

with the EM fields. In the microwave frequency range, only electrons are able to follow the fast

variations of the EM fields.

The microwave resonant cavity method is based on the theory of standing waves in cavities and

has been used for a long time for various discharges [128–131] and especially in dusty plasmas

[5, 132]. This method requires a metal box which serves simultaneously as reactor electrodes

and microwave cavity. When a highly symmetric configuration is chosen for experiments (i.e.

cylindrical or rectangular cavity), well-known EM modes can be excited (otherwise, the EM field

space shape has to be determined experimentally).

In a cylindrically symmetric cavity, the electric field vector−→E (−→x , t) can be written as:

−→E (−→x , t) =

−→E 0(x, y) exp(±ıkz − ıΩt) (3.15)

where both k and−→E 0(x, y) are determined by the cavity geometry. Generally, Ω is complex,

thus the imaginary part of ıΩt represents the oscillations and the real part of ıΩt represents the

64


damping of the mode. The dispersion relation is:

Ω2 = k2 c2

n2= k2 c2

µrǫr(3.16)

where c is the vacuum light velocity and n is the cavity medium refraction index. The relative

permeability of the medium µr is taken as µr = 1. The dielectric constant ǫr of the cavity

containing the plasma is EM field oscillation frequency dependant and is given by [5]:

ǫr = 1 + ı1

Q0+ ı

ω2pe

ω(νe − ıω)= 1 + ı

1

Q0−

ω2pe

ω2 + ν2e

+ ıω2

pe

ω2 + ν2e

νe

ω(3.17)

where ω is the excitation frequency, ωpe the electron plasma angular frequency and νe the

electron collision frequency. The second term ı/Q0 accounts for the non-ideality of the cavity

(finite conductivity of the walls, etc.). The third and fourth terms appear in the presence of

the discharge. The third term represents the frequency shift of the plasma compared to the

resonance frequency of the empty box and the fourth term represents the losses due to electron

collisions with other particles. It is obvious that the cavity method can be used as a diagnostic

since the electron frequency ωpe is directly related to the electron density ne (see Chap. 2.3.1).

If it is assumed that the deviations of ǫr from an ideal case are small (i.e. Q0 ≫ 1, ωpe/(ω2 +

ν2e )1/2 ≪ 1, etc.), Eq.3.17 can be substituted in the dispersion relation (Eq.3.16) and the result

can be linearised yielding for Ω the following:

Ω = ω0 + ∆ω − ıω0

2Q(3.18)

where ω0 is the resonance frequency without the plasma and ∆ω is the frequency shift caused

by the plasma. This frequency shift is:

∆ω =1

2ω0

ω2pe

ω2 + ν2e

(3.19)

Q is a quality factor of the cavity that accounts for the total dissipation:

1

Q=

1

Q0+

ω2pe

ω2 + ν2e

ν

ω0(3.20)

All experiments performed in this thesis were done at low pressure (P " 1 mbar). The electron

collisions are mainly with the neutrals and the electron collision frequency νe ≃ νen between

electrons and neutrals is νen = nnvTeσen ≃ 800 MHz where σen ≃ 5 · 10−20 m2 is the electron-

neutral collision cross section. As we operated with microwave frequencies around f = ω/2π ≃2700 MHz, the collision frequency ν can be neglected and using Eq.3.19, we obtain:

n0e =

2meǫ0ω2∆ω

e2ω0(3.21)

65


where n0e is the electric field average electron density. Indeed, electrons in the plasma are

generally not homogeneously distributed over the cavity and have a certain spatial profile ne(−→x )

giving:

n0e =

∫cavity ne(

−→x )E2(−→x )d(−→x )∫cavity E2(−→x )d(−→x )

(3.22)

If the spatial distribution of the electrons is needed, other diagnostics have to be used. However,

the density n0e obtained with Eq.3.21 has to be multiplied by a corrective factor A, accounting

for the spatial variation of the microwave field, which depends on the discharge geometry and

conditions. In dusty plasmas, the electron density is greatly influenced by the dust particles.

Hence, the corrective factor evolves with time. This factor is always close to unity [133] and for

this reason is taken as 1 in all our calculations .

Experimental procedure

plasma

generator

Microwave Oscilloscope

Matching BoxPower SourceRF

antennatransmitter

antennareceptive

Schottky diode

Figure 3.8: Principle schematic of microwave resonant cavity technique

Two antennas are fixed in the plasma box. One is powered by a microwave generator (Rhode

& Scharz signal generator with a 100 kHz-4320 MHz bandwidth) and the other is the receptor.

This last antenna is linked to a Schottky diode (Agilent Technologies) which converted the

amplitude of the microwave signal into a voltage measurable by an oscilloscope (Fig.3.8). By

scanning the plasma with frequency around the TM110 mode resonant frequency (2723 MHz

in the methane chamber), it is possible to reconstruct the resonance curve and to deduce the

resonance frequency and thus the electron density.

In order to be able to reconstruct a resonance curve, many experiments with the same parameters

were done with a different frequency applied to the emissive antenna by the microwave generator

for each plasma run. The repeatability of the experiments was checked by following the evolution

of the self bias voltage of the powered electrode. As the temporal resolution of the cavity is

about ∼ 100 ns, the output voltage was measured every 500 ns. Resonance curves can then be

reconstructed for both running discharge (Fig.3.9) and the plasma afterglow.

66


Figure 3.9: Resonance curves of the microwave cavity with or without plasma

3.2.6 Langmuir probe measurement

When a metal probe is inserted into a plasma, it will collect ion and electron currents. Depending

on the voltage applied on the probe tip, the drag current will be different as the ions and electrons

need to overcome different potential barriers. These probes, called Langmuir probes, have been

extensively studied by Mott-Smith and Langmuir [134].

For an applied probe potential equal to the floating potential Vf , there is no net current on

the probe as the electron and ion fluxes balance each other. For potentials below the floating

potential, the probe collects more ions than electrons and thus the current is, by convention,

negative and when the probe potential is highly negative all ions are able to reach the probe

surface and the current saturates to the ion saturation current. For probe potential above the

floating potential, the probe collects more electrons than ions and the current is, by convention,

positive. When the probe potential reaches the plasma potential Vp, ions and electrons are

collected with fluxes unaffected by the probe potential. By increasing the probe voltage beyond

Vp, all electrons are able to reach the probe and the current saturates to the electron saturation

current Iesat. Due to a higher mobility of the electrons, the electron current is much larger

than the absolute value of the ion saturation current Iisat. A typical current-voltage (I-V)

characteristic of a Langmuir probe is drawn in Fig.3.10.

Experimental procedure

In order to determine the evolution of the electron temperature of the plasma in the ComPLExS

reactor, Langmuir probe measurements were carried out. The Langmuir probe was a HIDEN

ESPion RF compensated probe with a 10 mm tungsten cylindrical tip with a diameter of 0.15

mm. The tip of the probe was placed near the edge of the electrodes (about 1 cm inside the

interelectrode volume) in the middle of the gap in between the electrodes (Fig.3.11). This

configuration was chosen in order to prevent large disturbances of the discharge due to the

presence of the probe (such as modifications of the self bias voltage behaviour or arcing to the

67


isat

VV Vf p

I

I

I

esat

Figure 3.10: Typical I-V characteristic of a Langmuir probe

Porthole

Porthole

Porthole

Argon inlet

Electrodes

ceramics stands

to computer

Electronic vacuum valve

To pumping system

Langmuir probe

Figure 3.11: Langmuir probe measurement schematic

68


probe tip).

First estimation

The first estimation that can be made from the probe measurement is the floating potential

with respect to the plasma potential and thus the electron temperature. Indeed it is known that

a potential drop across the presheath region is necessary to accelerate the ions to the Bohm

velocity. This potential drop is [20]:

Vp − Vs =kBTe

2e(3.23)

where Vp is the plasma potential and Vs is the potential at the sheath-presheath edge. The

potential drop within the sheath between a plasma and a floating wall can be determined by

equating the electron flux and the ion flux at the wall. One can obtain [20]:

Vf − Vs = −kBTe

eln

( mi

2πme

)(3.24)

where Vf is the floating wall potential. Combining Eqs. 3.23 and 3.24, the relation between the

electron temperature Te and the floating potential can be found:

Vf − Vp ≃ −kBTe

2eln

( mi

2.31 · me

)(3.25)

This method gives a first approximation of the electron temperature. However, it is not always

easy to obtain the plasma potential directly from the shape of the probe characteristic.

The EEDF procedure

When the gas pressure is around P = 1 Torr, it has been shown that the electron energy

distribution function (EEDF) is not Maxwellian [135]. It is still possible to determine the elec-

tron temperature using the second derivative of the probe current. Indeed, for non-maxwellian

electrons, the electron current is [20]:

Ie = eA

∫ ∞

−∞

dvx

∫ ∞

−∞

dvy

∫ ∞

vmin

dvzvzfe(v) (3.26)

where A is the probe area, fe(v) is the electron velocity distribution function and:

vmin =(2e(Vp − V )

me

)1/2(3.27)

where V is the probe voltage. Consequently the second derivative is [20]:

d2Ie

dV 2=

2πe3

m2Afe[v(V )] (3.28)

69


The EEDF is:

ge(ǫ)dǫ = 4πv2fe(v)dv (3.29)

where ǫ is the electron energy. Thus using Eqs.3.28 and 3.29, the EEDF can be derived from

the probe current second derivative:

ge(V ) =2me

e2A

(2eV

me

)1/2 d2Ie

dV 2(3.30)

Integrating the EEDF, the electron density ne can be found:

ne =

∫ ∞

0ge(ǫ)dǫ (3.31)

as well as an effective electron temperature:

Teff =2

3ne

∫ ∞

0ǫge(ǫ)dǫ (3.32)

This method is often used to study non-Maxwellian plasmas as it enables ne and Te to be

obtained without an assumption about the EEDF.

The Langmuir procedure

It is not always possible to derive properly the second derivative of the probe signal. Indeed,

the probe signal can be noisy and the noise is amplified by the differentiation procedure which

leads to non-usable second derivative characteristics.

It is known that for an ideal Maxwellian plasma, the electron temperature can be deduced di-

rectly from the electron part of the probe voltage-current characteristic. Indeed, for Maxwellian

plasmas, the logarithm of the electron probe current is [20]:

ln( Ie

Iesat

)=

e · (V − Vp)

kBTe(3.33)

In laboratory RF plasmas, the plasma is rarely Maxwellian and the EEDF varies with the

pressure from bi-Maxwellian at low pressure to Druyvesteyn-like for pressures around 1 Torr

[135] which was the pressure region of our experiments. It is nevertheless possible to obtain

a good approximation of the electron temperature and electron density using the Langmuir

procedure. Using the semilog plot of the probe characteristic, one can extract two linear parts

of the function ln(Ie(V )) where Ie is the electron current. The first one near the floating potential

(zero current crossing of the probe characteristic) and one for the highest voltage applied. By

extrapolating those two lines, the crossing occurs at the plasma potential Vp and the slope of

the first one gives a good approximation of the electron temperature. The plasma density is

then obtained from the Langmuir formula [135]:

I(Vp) = eAne

( kbTe

2πme

)1/2(3.34)

70


It has been shown that the results given by the Langmuir procedure on Druyvesteyn-like plasmas

underestimate the effective electron temperature by about 60 % and the plasma density is

overestimated by approximatively 55 % [135]. The order of magnitude are however in good

agreement and the Langmuir procedure may be used to obtain the relative changes in parameters.

Using the ion part of the I-V probe characteristic

The ion part of the Langmuir probe characteristic is commonly used in plasma diagnostics.

For example, the calculation of the plasma density from the ion current can be done in various

way. The OML approach is the most popular one as it does not require the knowledge of the

electron temperature and plasma potential, and in many experiments, the square of the ion

current I2i is found to be a linear function of the probe voltage as is expected by the theory.

However, the ion-ion and ion-atom collisions as well as the finite length of the probe affect the

orbital motion of the ions and tend to destroy it. The ion current under such conditions is due

to radial motion rather than orbital motion. The OML theory is already not valid for very low

pressure (P ! 0.02 Torr). Radial motion theory of Allen, Boyd and Reynolds (ABR) can be

also used to determine the plasma density but are difficult to use as it requires the knowledge

of the probe sheath voltage and an inference on the plasma density must done from the ion

current and the floating potential which requires a heavy iteration procedure. Consequently,

this method is mostly used in a confirmatory mode rather than a predictive mode. For pressure

around 1 Torr, the OML theory tends to overestimate the ion density while the ABR theory

tends to underestimate it (see Ref.[135] and references therein for more details).

In our experiments, the ion density was deduced from the ion part of the probe current using

the software furnished by HIDEN analytical which enables analysis using both the OML and

ABR theory.

3.2.7 Electron microscopy analysis

The direct observation of the dust particles is, when possible, the best way to obtain their size

and shape. Hence, dust particles grown in the Methane and ComPLExS RF discharges have been

collected and analysed ex-situ by electron microscopy. The dust particles grown in the methane

reactor have been analysed using both scanning electron microscopy (SEM) and transmission

electron microscopy (TEM) while the dust particle grown in the ComPLExS reactor have been

analysed using only SEM.

Scanning electron microscopy

For SEM, the sample is scanned with a high energy electron beam (several keV). The electrons

interact with the atoms of the sample that produce signals containing information about the

surface topography, composition, electrical conductivity, etc. The most common technique for

imaging with SEM is the collection of secondary electrons emitted by the sample. Because the

mean free path of these electrons through matter is very short, only those emitted close to the

71


surface of the sample can be collected and thus, high resolution images of the sample surface

can be reconstructed (down to 1 nm).

Transmission electron microscopy

For TEM, a beam of electrons focused with electromagnetic lenses is transmitted through an

ultrathin specimen and interacts with it. Depending on the density of the material, some

electrons are scattered and disappear from the beam. The unscattered electrons hit a fluorescent

screen at the bottom of the microscope. A“shadow” image of the specimen is created with the

darkness of its different parts related to the opacity of the corresponding regions in the specimen.

The most used methods to display the image of the sample are the “dark field mode” and the

“bright filed mode”. For the dark field mode, the image is creating from a diffracted electron

beam. It is particularly useful for studying the crystalline structure of the sample as, depending

on the crystal orientation, the electrons pass through the sample in a straight line and one obtain

a bright contrast or are deviated and one obtains a dark contrast. For the bright field mode, the

screen is placed in the image plane and only the unscattered transmitted electrons are collected.

A magnified image of the sample is then observed.

3.3 Conclusion

In this chapter, the different plasma reactors used for this thesis were described. The PKE-

Nefedov reactor was used for the studies of all the different phases of a RF discharge life. The

Methane reactor was used as a reference for stutying dust particle growth and for the studies

of the electron density decay in afterglow plasmas due to the limitation of the PKE-Nefedov

chamber. The ComPLExS reactor was used to investigate the dust particle growth by RF

sputtering and its effect to the discharge and plasma parameters. The ComPLExS results were

then compared to the PKE-Nefedov results to extract essential features of the influence of dust

growth by RF sputtering.

The wide range of diagnostics presented in this chapter were used in order to correlate as far as

possible the different data and extract important physical information.

72



Dans ce chapitre, les differents dispositifs experimentaux utilises durant cette these sontpresentes.

– Le premier reacteur presente est le reacteur PKE-Nefedov (Fig.3.1). C’est une dechargeRF capacitive developpee pour des experiences en microgravite. La chambre a vide est unparallelepipede de verre de 10×10×5 cm3 de volume. Les electrodes sont des disques d’acierinoxydable de 4.2 cm de diametre polarisees en mode “push-pull”. L’espace interelectrodeest de 3 cm. La pression d’argon de travail est entre 0.1 et 2 mbar et la puissance maximaleinjectee dans le plasma est de 4 W. Ce reacteur a ete utilise pour tous les themes abordesdans cette these.

– Le second reacteur est le reacteur a dilution de methane. C’est une boıte a decharge enacier inoxydable enfermee dans une chambre a vide (Fig.3.2). La boıte a decharge estun cylindre creux de 13 cm de diametre interieur relie a la masse (l’anode). Le base estfermee par une grille percee de trous de 1 mm de diametre et ayant une transparence de20 % afin de travailler avec un ecoulement de gaz laminaire et uniforme. L’electrode RFest en forme de pomme de douche en acier inoxydable qui vient fermer le haut de la boıtea decharge. Elle a un diametre de 12.8 cm et une epaisseur de 1 cm et est fermee parune grille similaire a celle de l’anode. Un tube en acier inoxydable assure l’alimentationRF et l’injection du gaz. La puissance varie entre 0 W et 100 W. Les gaz sont injectespar deux lignes independantes : une pour l’argon (ou l’azote) avec un flux maximum de45 sccm, et une pour le methane (ou le silane) avec un flux maximum de 10 sccm. Lapression de travail varie entre 0.1 mbar et 1.5 mbar. Ce reacteur a ete utilise pour lesrecherches concernant la croissance de poudres et celles concernant la decroissance de ladensite electronique durant la phase post-decharge.

– Le troisieme et dernier reacteur est le reacteur ComPLExS (Fig.3.3). Il est constitue dedeux electrodes paralleles enfermes dans une chambre a vide cylindrique en acier inoxydablereliee a la masse de 30 cm de diametre et de 30 cm de hauteur. L’electrode inferieure estpolarisee a la RF. Elle est en aluminium et a un diametre de 10 cm. L’electrode superieureest un disque en aluminium de 11.5 cm de diametre relie a la masse et pose sur 3 plots deceramique. L’espace interelectrode est de 2 cm. La pression de travail varie entre 0.1 Torret 2 Torr. L’argon est injecte avec un flux maximum de 20 sccm. Le tension RF crete acrete peut atteindre 350 V. Ce reacteur a ete utilise pour etudier la croissance de poudrespar pulverisation.

Differents diagnostics ont ete utilises pour analyser les differents parametres des decharges, desplasmas et des poudres :

– Diagnostics electriques : tension d’autopolarisation (PKE-Nefedov, reacteur methane, Com-PLExS) et amplitude de l’harmonique fondamental du courant RF (PKE-Nefedov).

– Diagnostics optiques (PKE-Nefedov) : imagerie video (classique et rapide), luminosite duplasma.

– Spectroscopie d’emission (ComPLExS).– Cavite resonnante a microondes (reacteur methane).– Sonde de Langmuir (ComPLExS).– Analyse par microscopie electronique (reacteur methane et ComPLExS).

La correlation entre les resultats obtenus grace ces differents diagnostics est utilisee pour extraireun maximum d’informations sur les processus physiques mis en jeu.

73

Chapter 4

Dust particle growth and related

instabilities in argon RF discharges

In the laboratory, dense clouds of submicron particles light enough to fill the gap between the

electrodes can be obtained using reactive gases such as silane [34, 119] or using a target sputtered

with the ions coming from the plasma [59–61, 73, 74]. During the dust particle growth, plasma

instabilities sometimes occur.

In the first part of this chapter, the growth of dust particles in sputtering discharges is discussed

and the modification of the plasma parameters due to the presence of the growing dust particles

is discussed. In the second part of this chapter, instabilities occuring during dust particle growth

are investigated experimentally.

4.1 Dust particle growth in plasmas

In chemically active discharges such as Ar/SiH4 or Ar/CH4 plasmas, the growth of dust par-

ticles occurs naturally. The growth speed is more or less greater depending on the relative

proportions of the gas components [35, 36, 119]. It is known that dust particle growth in a

plasma modifies the discharge properties and therefore the dust particle growth kinetic can be

monitored looking at the evolution of discharge parameters such as the current amplitude (and

its harmonics) or the self bias [119, 123]. The growth of dust particles in chemically active

discharges has been actively studied. Growth occurs in four phases:

1. Molecular precursors coming from gas dissociation

2. Formation and accumulation of nanoparticles from these precursors

3. Agglomeration of the nanocrystallites

4. Growth by molecular sticking

In sputtering discharges, dust growth has been studied for different materials [59–61, 73, 74, 136].

These results suggest that the growth mechanism is roughly the same except that the molecular

74

4.1 CHAPTER 4. DUST PARTICLE GROWTH AND INSTABILITIES

precursors come from the sputtered materials. The number of studies devoted to dust particle

growth by RF sputtering is however smaller than the number of studies devoted to dust particle

growth in chemically active discharges.

Consequently, we decided to study the growth of dust particles in sputtering discharges. The

sputtered material was melamine-formaldehyde (MF) coming from previously injected dust par-

ticles inside the discharge. MF is a carbonaceous polymer and preliminary studies showed that

the growing dust particles are mainly composed of carbon [66]. In the next section, the growth

of particles by RF sputtering and its effects on the discharge are studied.

4.1.1 Dust particle growth in Ar/CH4 plasma

0 5 10 15 20 25 30 35 40 45

−10

−8

−6

−4

−2

0

Self bias evolution during dust growth in Ar/CH4 discharge

Time (s)

Vdc (

V)

Figure 4.1: Evolution of the self-bias during the growth of dust particles in Ar/CH4 plasma

In order to have a reference for the studies of the dust particle growth by RF sputtering, the

growth of dust particles in an Ar/CH4 discharge has thus been investigated. The measurement

was made in the methane reactor (see Sec.3.1.2). The argon flow was 41.8 sccm and the methane

flow was 0.8 sccm (the methane dilution rate was 1.9%). The operating pressure P was 1.1 mbar

and the RF power was PW = 20 W. The evolution of the self bias voltage on the driven electrode

during the particle growth has been recorded and is presented in Fig.4.1. It shows that during

the growth process, the self-bias voltage at the cathode gradually increases. After an initial

period of little change a steady increase in self-bias voltage (a decrease in |Vdc|) was observed

with an asymptotic approach to an apparent plateau value. As the plasma changes, the self-

bias voltage achieves a value which ensures that no direct current is carried by the powered

electrode [20, 137]. Qualitatively, the effect of particle growth on the self-bias can be seen as

a consequence of the reduction in electron density due to negative charge accumulating on the

particles: a reduced self-bias will reduce the ion flux and allow a longer period of electron flow

so that a new equilibrium is established. This result is in agreement with the results obtained

in Ar/SiH4 plasma [119] in which the self-bias voltage evolution was correlated to the dust

growth process. After 43 s, the discharge was stopped and the dust particles were collected for

75


analysis. SEM and TEM measurements show that the grown dust particles are spherical and

quasi monosized (rd ∼ 200 nm; see Fig.4.2).

(a) TEM image (b) SEM image

Figure 4.2: Dust particles grown in a Ar/CH4 plasma

4.1.2 Dust particle growth in the PKE-Nefedov reactor

The dust particles were grown by sputtering of a polymer layer deposited on the electrodes and

coming from previously injected dust particles (MF dust particles). They were grown following

this routine. First, the plasma chamber was pumped down to the lowest possible pressure (base

pressure ∼ 2 · 10−6 mbar). Then, argon was injected in the reactor up to the working pressure

(1.2 mbar≤ P ≤ 1.6 mbar static pressure (no gas flow)) and the discharge is ignited. The input

power was 3 W < PW < 4 W. After few tens of seconds, a dust particle cloud appeared in the

plasma and was visible by naked eye due to the scattered laser light from the dust particles.

Various situations can be obtained after the dust particle growth (several layers, domelike shaped

cloud, 3D dense cloud). This is due to the fact that the sizes and densities of the grown dust

particles can not be controlled precisely. Indeed, it has been shown that the growth process in

the PKE-Nefedov reactor is extremely sensitive to gas purity [61, 66]. The purity of the gas was

also reported to be an important requirement for dust particle growth in similar experiments

[59]. As our experiments were performed without gas flow, this effect was amplified. It was

indeed necessary to pump for at least 1 hour between each experiments in order to be able to

grow dust particles again: the species formed during the previous run and/or coming from the

outgassing of the walls and/or from the sputtered matter needed to be eliminated. This was

confirmed using plasma emission spectroscopy on successive runs [66]. Impurities such as OH

or N2 were observed and their influence is still under investigation. Moreover, possible dust

particle precursors like carbon molecules C2, CN , and CH appeared. The C2 molecule seems

to be a good indicator of dust formation. High dust density was observed only when the base

pressure was sufficiently low (less than 10−5 mbar).

76


Electrical measurements

Figure 4.3: a) Self-bias (Vdc) of the bottom electrode b)Amplitude of the fundamental harmonicsof the RF current (P=1.6 mbar and PW =3.25 W)

The self-bias voltage of the bottom powered electrode of the PKE-Nefedov chamber has been

recorded during the dust particle growth process. As can be seen in Fig.4.3(a), the self bias in-

creases during the dust particle growth (or decreases in absolute value) after a small period of

time just after plasma ignition where it decreases slightly. This behaviour is almost equivalent

to that observed in chemically active discharges.

The appearance of dust particles in the plasma can also be confirmed by recording the time

evolution of the amplitude of the fundamental harmonic of the RF current. Indeed, as dust par-

ticles appear and grow in the plasma, they acquire more and more electrons and the amplitude

of the current decreases as a consequence. In Fig.4.3(b), this effect is clearly seen: the current

amplitude decreases continually after a short period of little change just after plasma ignition.

This diagnostic seems to be slightly more sensitive than the self bias voltage.

During dust particle growth, instabilities can appear on the electrical signals (self-bias volt-

age and amplitude of the first harmonic of the discharge current): the dust particle growth

instabilities (DPGI). The DPGI are observed only when the base pressure is below 10−6 mbar

which confirms that gas purity is an important parameter affecting the dust particle growth pro-

cess. Instabilities during dust particle growth were also reported in other sputtering discharges

[59, 73, 74]. A detailed study of this instability is presented in Sec.4.2.

It is generally believed that the DPGI are directly linked to size and density of the dust particles

in the plasma [59, 74]. Therefore, this instability can be used to study the dust particle growth

kinetic.

The DPGI typically appear tens of seconds after plasma ignition. The beginning of the DPGI

is characterised by strong amplitude oscillations in both the amplitude of the fundamental har-

monic of the discharge current and the self-bias voltage of the lower electrode (Fig.4.3). By

77


looking at the plasma glow with the naked eye, the beginning of the DPGI can also be detected

since the light emitted by the plasma is strongly affected.

The DPGI starting time changes slightly from one experiment to another due to differences in

the dust particle density which strongly depends on gas purity. Nevertheless, statistics per-

formed on many experiments allow some general behaviour to be extracted. It is shown that the

appearance time of the DPGI, and consequently the dust particle growth rate, depends on the

gas pressure: the higher the pressure, the shorter the appearance time. Indeed, as can be seen in

Fig.4.4, the DPGI appearance time decreases while increasing the pressure. In previous exper-

iments, it was found that a threshold pressure for dust particle growth exists around 1.2 mbar

[66]. Thus, for the same input power PW =3.25 W, nearly 2 minutes are required to initiate the

DPGI at 1.4 mbar while only 40 s are necessary at 1.8 mbar. Two hypotheses can explain this

behaviour: either a longer time is required to reach the critical dust density necessary to cause

the DPGI at low pressure, or the dust density that could be reach at low pressure is lower and

bigger dust particles are needed to initiate DPGI (i.e. longer time). However, depending on the

initial conditions, two regimes of DPGI exist: a slow one and a fast one. These regimes have

different appearance times. Therefore, the dependence on the input power is not so clear even

if it seems that the higher the input power, the shorter the appearance time (Fig.4.4).

Slow regime

1.4 1.50 1.60 1.70 1.80 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

Ap

pea

ran

ce t

ime

(s)

Ap

pea

ran

ce t

ime

(s)

40

60

80

100

120

40

60

80

Pressure (mbar) Power (W)

a) b)

Fast regime

Figure 4.4: Appearance time (a) as a function of pressure (b) as a function of the input power.

Laser extinction measurement

In order to know when the dust particles reach a detectable size and/or density, laser extinction

measurement has been performed. The evolution of the transmitted laser light intensity is shown

in Fig.4.5. As can be seen the current amplitude has already started to decrease indicating that

some dust particles are already formed in the plasma while the transmitted laser light is still

at its maximum value. It indicates that either the density or the size of the dust particles is

very small. However, in chemically active discharges, the dust density is very high when the

dust particles start to grow [36]. It is probably the same in this experiment. Around 50 s,

the transmitted laser light starts to decrease indicating that the dust particles are now large

78


enough to affect the laser light transmission (the scattered light increases as a function of the

dust particle radius; Sec.3.2.3). It is worth noting that this decrease occurs before the DPGI

start: the plasma is filled with detectable dust particles before the ignition of the DPGI. It may

be an indication that the agglomeration phase occurs before the DPGI start. The transmitted

laser light steadily decreases showing that dust particles grow continually in the plasma

However, the PKE-Nefedov reactor does not allow us to collect samples of the grown dust

particles easily as this reactor was originally designed for microgravity experiments and thus

no easy access for sample collection exists. Consequently, it is not possible to deduce the dust

sizes and/or the dust density inside the reactor. Moreover, the exact optical index of the dust

particles is not known as it is dependent on the material and the exact chemical composition

of the dust particles is not known even if it has been proven that the dust particles are mainly

composed of carbon [61].

Figure 4.5: Current amplitude (blue curve) and relative laser intensity (black curve) as a functionof time during dust particle growth (P =1.6 mbar and PW =3.25 W).

Plasma emission spectroscopy

Plasma emission spectroscopy was performed during the dust particle growth. The evolution of

the intensities of several argon lines was monitored. The evolution of the 750.39 nm argon line

is presented in Fig.4.6(a). As can be seen, the line intensity increases during the particle growth

process. If we consider that the energy level from which the transition occurs is principally

populated by direct excitation from the ground state1 the line intensity increase should indicate

that the electron temperature increases (see Sec.3.2.4). This result would be in agreement with

the results obtained by Bouchoule and Boufendi in a Ar/SiH4 discharge where they reported

that the electron temperature increases when dust particles are growing inside the plasma [33].

It has also been shown that the presence of dust particles can increase the metastable density

in the plasma [33]. The line intensity ratio I763.5/I800.6 of the 763.5 nm and 800.6 nm argon

1For levels of interest corona equilibrium requires ne << 1011 cm−3 [126]. In our experiments, ne ∼ 108−

109 cm−3

79


0 50 100 150 2000

20

40

60

80

100

120

Time (s)

Inte

nsi

ty (

a.u

.)

Exp. data

quadratic

(a)

0 20 40 60 80 100 120 140 160 180 2001.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

Time (s)

Ra

tio

I763.5

/I800.6

smooth data

raw data

(b)

Figure 4.6: a) Evolution of the intensity of the 750.39 nm argon line during particle growthin the PKE-Nefedov reactor with pressure P = 1.6 mbar and input power PW = 3.25 W. b)Evolution of the Intensity ratio I763.5/I800.6 of argon lines during particle growth for a pressureof P = 1.6 mbar and an input power PW = 3.3 W

lines have been recorded during the dust particle growth (Fig.4.6(b)). As can be seen, the ratio

is lower when dust particles are present in the plasma than when the plasma has just been

ignited (pristine conditions). It indicates that the 3P2 argon metastable density increases when

dust particles are growing in the plasma. Indeed, if the electron density decreases during dust

particle growth and the electron temperature (and thus the electron energy) increases, the net

rate of excitation of the argon excited levels is increased. The metastable concentration increases

because the quenching rate by electron collision decreases and the quenching by collision with

dust particles is thought to be neglectable. This effect was already observed by Boufendi in

Ar/SiH4 mixture plasmas [36]. However, as can be seen, the ratio slightly increases after it has

reached a minimum a few tens of seconds after plasma ignition indicating that the metastable

density tends to diminish. The value of the ratio is still below the value it had just after plasma

ignition meaning that the metastable density remains higher than in a dust-free plasma.

4.1.3 Dust particle growth in the ComPLExS reactor

The growth of dust particles by RF sputtering from MF dust particles deposited on the elec-

trodes was also performed in the ComPLExS reactor following the following routine. Firstly, the

plasma chamber was pumped down to the lowest possible pressure (base pressure ∼ 5·10−6 Torr).

Then, argon was injected into the reactor up to the working pressure (0.6 Torr < P < 1.8 Torr

with a flow of 17.9 sccm) and the discharge was turned on. The peak to peak RF voltage was

230 V < Vpp < 320 V. After a few tens of seconds, a dust particle cloud appeared in the plasma

and was visible to the naked eye as a result of the laser beam light scattered by the dust particles.

80


! "! #! $! %! &!!!'#

!!'"&

!!'"

!!'!&

!

Time (s)

Vd

c (

x1

00

V)

Figure 4.7: Evolution of the self bias voltage of the powered electrode in the ComPLExS reactorduring dust particle growth (Pressure P=1.75 Torr and RF voltage Vpp=280 V)

Self bias voltage measurement

As in the PKE-Nefedov reactor, the growth of dust particles can be monitored by following some

electrical characteristics of the discharge. The evolution of the self-bias voltage of the powered

electrode during the dust particle growth is shown in Fig.4.7. As can be seen, an increase of

the self-bias Vdc (a decrease in absolute value) is observed during the growth cycle after a small

period of little change. It is similar to the behaviour observed in the PKE-Nefedov reactor. The

total self-bias voltage change is about 50-60 % over a one minute discharge run. A decrease of

the RF current amplitude of about 50-60% and a small increase of the RF peak to peak voltage

of about 30% are also observed (not presented here). This is due to the modification of the

discharge conditions (and therefore the matching conditions) caused by the presence of the dust

particles.

Studies of the dependence of the evolution of the self bias voltage (and thus the dust particle

0.6 0.8 1 1.2 1.4 1.6 1.80

10

20

30

40

50

60

70

Pressure (Torr)

Tim

e (

Sec)

Regimen A

Fit

Fit

Regimen B

240 250 260 270 280 290 300 3106

8

10

12

14

16

18

20

22

Vpp

(V)

Tim

e (s

)

Figure 4.8: Self-bias increase time as a function of: (left) pressure; (right) RF peak to peakvoltage at the start of the plasma run (Regimen B)

81


growth kinetic) on different experimental conditions have been carried out. In Fig.4.8, the re-

sults of studies as a function of the argon pressure P and the peak to peak RF voltage Vpp are

presented. It shows the time needed for the self bias to increase by 5% from its minimum value

corresponding to the dust-free plasma condition just after plasma ignition.

An impurity effect was observed in the ComPLExS reactor. Indeed when the pumping time

between experiments is one hour (regimen B presented in Fig.4.8) the 5% Vdc increase time

is much longer than when the pumping time in between the experiments is around 12 hours

(regimen A in Fig.4.8). It indicates that the dust particle growth is delayed when the pumping

time is shorter. This may be due to the presence of impurities such as residual nitrogen or

OH compounds in the plasma chamber. As previously mentioned, a similar impurity effect was

reported in the PKE-Nefedov chamber [66].

In Fig.4.8, it can be clearly seen that the argon pressure plays a major role in the dust particle

growth process. Indeed, the time needed for the 5% Vdc increase is reduced by increasing the

pressure. Below a threshold pressure (around 0.5 Torr in the ComPLExS reactor operating in

regimen B) no dust growth is observed over plasma runs of duration of a few minutes (typically

up to 10 minutes). The dependence on the RF voltage is not as clear: as for the lowest Vpp, it

seems that the 5% Vdc increase takes longer than for higher values, but no clear trend can be

extracted from the data in Fig.4.8.

Correlation of the dust size with the self bias measurement

Different dust samples have been collected on silicon wafers for different discharge durations.

The wafers were placed on the powered electrode. The argon pressure was P =1.75 Torr and

the peak to peak RF voltage was Vpp =280 V at plasma ignition. Each sample accumulated the

grown dust particles over a minimum of 20 plasma runs. Then, SEM images of these samples

have been taken. The sizes of the dust particles have been measured. The results for different

discharge durations are:

1. First sample (Fig.4.9(a)): discharge duration of t ∼ 10 s (the discharge was switched off

just before the fast increase of the self bias voltage; see Fig.4.7). The mean radius of the

collected particle is 16±5 nm

2. Second sample (Fig.4.9(b)): discharge duration of t ∼ 20 s (the discharge was switched off

at the end of the fast increase of the self bias voltage; see Fig.4.7). The mean radius of

the collected particle is 24±6 nm

3. Third sample (Fig.4.9(c)): discharge duration of t ∼ 30 s. The mean radius of the collected

particle is 33±8 nm

4. Fourth sample (Fig.4.9(d)): running time of t > 120 s. The mean radius of the collected

particle is about 100 nm. However, there is a large dispersion. The possible explanation

is the existence of many generations of dust particles.

82


(a) ton = 10 s (b) ton = 20 s

(c) ton = 30 s (d) ton > 120 s

Figure 4.9: SEM images of the dust particles collected in the ComPLExS reactor for differentdischarge times.

The particle were collected over many runs on the powered electrodes. As a result, it may be

possible that the real size of the dust particle is slightly larger due to a possible re-sputtering of

the grown dust particles of the first runs. However, note that the fast variation of the self bias

between 10 s and 20 s is not associated with a sudden variation of the dust particle size. Indeed,

the size augmentation between samples 1 and 2 is nearly the same as between samples 2 and 3.

Spectroscopic measurement

Spectroscopic measurements were performed during the dust particle growth process. The argon

pressure was P =1.75 Torr and the peak to peak RF voltage was 280 V. The evolution of the

intensities of three argon lines has been monitored (415.86 nm, 518.77 nm, 750.38 nm) and

different line ratios have been calculated (Fig.4.10). As can be seen the intensity of each line

increases with time as in the PKE-Nefedov reactor. However, the ratios I415/I750 and I518/I750

firstly decrease before slightly increasing again. The interpretation of spectral measurements in

terms of plasma parameters presents some complications. The most general approach is to use

a collisional-radiative model which takes account of all collisional and radiative processes that

influence the populations of excited states, and hence the intensities of spectral emission lines

(as a function of the pressure, electron density and electron temperature) (see Sec.3.2.4). In

order to simplify the problem the electron distribution is invariably assumed to be Maxwellian.

Low pressure discharges such as the one used in this work allow a simplified version of the model

83


0 10 20 30 40 5010

−1

100

101

102

Time (s)

Inte

nsi

ty (

a.u

.)

750.3

518.77

415.88

(a)

0 10 20 30 40 5010

−3

10−2

10−1

100

Time (s)

lin

e r

ati

o

518.77/750.39

415.88/750.39

(b)

Figure 4.10: (a) Line intensities and (b) line ratio during dust particle growth at P = 1.75 Torrand Vpp = 280 V

0 10 20 30 40 500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Te (

a.u

.)

from I415.86

/I750.39

from I518.77

/I750.39

Figure 4.11: Evolution of the electron temperature Te estimated from the evolution of the lineratio

84


to be used: the corona approximation, in which the dominant processes are collisional excitation

from the ground state and radiative de-excitation.

The rising spectral line intensities during the particle growth process (Fig.4.10(a)) (when electron

density is falling) implies an increase in electron temperature. In the latter circumstances the

corona model predicts that the intensity ratios of Fig.4.10(b) should rise rather than fall (which

implies a decrease of the electron temperature as can be seen in Fig.4.11). The resolution of

this discrepancy will need to take account of the well-known fact that such discharges have

non-Maxwellian distributions: at sufficiently low pressures the distribution is bi-Maxwellian,

becoming Druyvesteyn-like at higher pressures [138, 139]. The present experiment falls within

the latter regime. The effect of large dust concentrations on the electron distribution is however

problematic. It can be supposed that the fast electrons of the EEDF are the only ones that can

overcome the potential barrier of the dust particles at floating potential and are thus collected

by them leading to a lack of energetic electrons which might be compatible with a decrease of

the line intensity ratios. It might also explained why the metastable density decreases slightly

after a few tens of seconds as observed in the PKE-Nefedov reactor. A recent kinetic model

suggests that the Druyvesteyn-like distribution is modified by the presence of dust particles in

such a way that it becomes nearly Maxwellian [140]. The slow increase in the intensity ratios

after 20 s could be interpreted as a providing support for this prediction.

Langmuir probe measurement

Measurement in a pristine plasma:

!!" !#" !$" " $" #" !"

"

#

%

&

'

$"

$#

$%

$&

$'

#"()$"

!%

V (V)

I (A

)

(a)

! " #! #" $! $" %! %"

!

#!

$!&'#!

!"

V (V)

dI/

dV

(A

/V)

'

'

! " #! #" $! $" %! %"!#

!

#

$&'#!

!"

V(V)

d2I/

dV

2 (

A/V

2)

(b)

0 5 10 150

0.02

0.04

0.06

0.08

0.1

Energy (eV)

EE

DF

(a.u

.)

(c)

Figure 4.12: (a) Langmuir probe characteristic. The vertical arrow indicates the position of theplasma potential. (b) First and second derivatives of the probe signal. (c) Estimated EEDFfrom the second derivative of the probe characteristic (argon plasma with P = 0.3 Torr andVpp = 240 V).

Langmuir probe measurements have been performed in an argon plasma with parameters

which do not favour the dust growth but close to conditions in which dusty plasma experiments

were performed. The I-V Langmuir characteristic is presented in Fig.4.12(a) for an argon plasma

with P = 0.3 Torr and Vpp = 240 V. The plasma potential Vp corresponds to a change of slope

85


in the I-V curve and is indicated by a vertical arrow. The position of Vp can be better estimated

on the 1st and 2nd derivative curves (Fig.4.12(b)). The derivatives have been computed after a

Savitzky-Golay filtering (polynomial filtering) of the probe signal in order to reduce noise. The

plasma potential corresponds to the first maximum of the 1st derivative and the zero-crossing of

the 2nd derivative. From the second derivative, it is possible to estimate the EEDF (Fig.4.12(c)).

As can be seen, the EEDF is far from a Maxwellian distribution and is Druyvesteyn-like. Simi-

lar results have been previously reported in other RF argon discharges with similar parameters

[135]. However, the energetic tail of the EEDF should decrease faster as electrons with energy

close to the ionisation level are lost in excitation and ionisation processes. This is due to the

noise in the probe signal. It is still possible to extract some plasma parameters using the second

derivative analysis: the electron density is ne ≃ 2.9 · 109 cm−3 and the effective electron tem-

perature Teff ≃ 5.2 eV (this last parameters is slightly overestimated due to the noise-induced

increase of the energetic tail of the EEDF).

Measurements in a dusty plasma:

!!" !#" " #" !" $"!!

"

!

%

&

'

#"()#"

!%

Probe potential (V)

Cu

rren

t (A

)

)

)

*)+)%),

*)+)$%),

Figure 4.13: I-V Langmuir probe characteristics at different instants of the discharge life (P =1.75 Torr and Vpp = 280 V). The curves have been smoothed.

The Langmuir measurements in a dust-free plasma with parameters close to the parameters

of dusty plasma experiments have confirmed that the EEDF is not Maxwellian. However, in

order to complement spectroscopic data, Langmuir probe measurements have been performed

during the dust particle growth in order to estimate the electron temperature. Some I-V probe

characteristics are shown in Fig.4.13 just after plasma ignition and at a discharge time t >30 s,

after the fast increase of the self bias voltage. Due to very noisy data, the second derivative

analysis method could not be applied and classical Langmuir analysis of probe characteristic

has been performed.

The ion density was deduced from the ion current part the probe characteristics and is found to

be quasi-constant during the whole duration of the discharge (ni ≃ (1.3±0.5)·109 cm−3 using the

86


Allen-Boyd-Reynolds (ABR) analysis and ni ≃ (1.3±0.4) ·1010 cm−3 using OML analysis2). As

can be seen in Fig.4.13, the slope of the electron current (positive part of the curves) decreases

after plasma ignition (the slope at t = 4 s is larger than the slope at t = 30 s). This is a strong

indication that the effective electron temperature increases during the dust particle growth. As

can be seen in Fig.4.14(a), the absolute value of the floating potential (Vp−Vf ) is increasing with

time which supports an increase of the electron temperature. The effective electron temperature

elevation is confirmed in Fig.4.14(b): it shows that the effective electron temperature rises from

2.5 eV at plasma ignition to about 3.5 eV after 50 s. As can be seen the effective electron

temperature elevation is more pronounced around 10 s after plasma ignition which corresponds

to the fast increase of the self bias voltage.

0 10 20 30 40 50

0

5

10

15

2.5

7.5

12.5

!"#$%&'

()!(*+%('

(a)

0 10 20 30 40 500

5

1

2

3

4

Time (s)

Te (

eV

)

(b)

Figure 4.14: Evolution of plasma parameters obtained with Langmuir probe measurements inan argon plasma with P = 1.75 Torr and Vpp = 280 V. (a) Evolution of the plasma potentialand the floating potential extracted from I-V curves. (b) Evolution of the electron temperaturededuced from the I-V curves.

Finally, from both spectroscopic and Langmuir probe measurements, it can be deduced that

dust particles in the plasma induce an increase of the effective electron temperature. However,

as the intensity of selected argon lines increases with time while the line ratio decreases, it tends

to show that the energetic electrons of the EEDF tail may be lost on dust particles.

4.1.4 Comparison of results

The growth of dust particles has been investigated in both the PKE-Nefedov reactor and the

ComPLExS reactor. Some general behaviour can be extracted. The most interesting features

are:

2It is known that the first method tends to underestimate the ion density while the second one tends tooverestimate it in our range of pressures [135].

87


• The growth of particles in the plasma modifies the electrical characteristics of the discharge.

The current amplitude and the absolute value of the self-bias voltage decrease during the

particle growth. This is due to the presence of dust particles which affects the electrons.

The electron density decreases due to electron attachment onto the dust particles which

is thought to be the main reason for both the decrease of the self bias voltage and current

amplitude.

• Spectroscopic measurements in both discharges show a rise in the absolute intensity of

some argon lines. As the electron density decreases during the dust growth, it indicates

that the electron temperature increases. However, comparison of the evolution of the

relative intensity of some argon lines in the ComPLExS reactor suggests that the electron

temperature would decrease in the case of a Maxwellian EEDF. This is a strong indication

that the EEDF is not Maxwellian and the ratio falls can be due to a depletion of energetic

electrons.

• Langmuir probe measurements confirm that the EEDF is not Maxwellian and that the

effective electron temperature increases during dust particle growth. Comparison with

the spectroscopic measurements supports the idea that there could be a depletion of the

energetic electrons.

4.2 Dust particle growth instabilities

In argon-silane RF discharges, instabilities have been observed during the growth of nanopar-

ticles on both the amplitude of the third harmonic (40.68 MHz) of the discharge current and

the self-bias voltage (Vdc) [119]. A proposed explanation was an attachment induced ionization

instability as observed in electronegative plasmas. Instabilities during dust particle growth have

also been observed in sputtering discharges [59, 73, 74]. These instabilities were divided into two

modes appearing consecutively: the filamentary mode and the great void mode. The triggering

of the instabilities was explained by spontaneous local fluctuations of the dust number density

leading to a lower depletion of the electrons in the regions of reduced dust particle density induc-

ing a higher local ionisation rate. It creates two forces on the negatively charged dust particles:

an inward electric force induced by the positive space charge (and thus electric field) and an

outward ion drag force. The threshold for the instability is determined by particle size (charge)

and the electric field strength.

4.2.1 Instabilities in the PKE-Nefedov chamber

Similar instabilities have also been observed during the dust particle growth in the PKE-Nefedov

reactor [141] but the separation in modes was more complicated. In this section, we focus on

the studies of the dust particle growth instabilities at an argon pressure P = 1.6 mbar and an

input power PW = 3.25 W . Dust particle growth instabilities (DPGI) typically appear tens of

seconds after plasma ignition and can be well observed on both electrical and optical signals

88


[68, 75, 141]. It is found that the DPGI have two different regimes: a fast regime and a slow

regime. These two regimes are characterized by their frequencies and evolution kinetic.

Phases and regimes of the instability

Phases of the DPGI

As mentioned previously, DPGI are evidenced by strong modulations of current amplitude

and self bias voltage (see Fig.4.3). As can be seen in Fig.4.15, the DPGI are characterised by

different phases which can be better observed looking at the current amplitude evolution in AC

mode in order to improve the vertical resolution of the oscilloscope.

In Fig.4.15, the measurement of the AC component of the fundamental harmonic of the RF

current is presented with its corresponding Fourier spectrogram for the total duration of a

plasma run. The beginning of DPGI occurs approximatively 40 s after plasma ignition and is

0 100 200 300 400 500 600 700−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

Am

pli

tude

(a.u

.)

1 2

3

4

5

6

7

Time (s)

Fre

qu

ency

(H

z)

0 100 200 300 400 500 6000

100

200

300

400

500

600

Fre

quen

cy (

Hz)

Time (s)

Figure 4.15: Left: Current fundamental harmonic amplitude (AC component). Successive phasesare numbered from 1 to 7. Right: Corresponding spectrogram

detected through a strong variation of the AC component of the current amplitude. Clear phases

can be identified and are numbered from 1 to 7 in Fig.4.15. These phases are better evidenced

by looking at the Fourier spectrogram of the signal. In order to highlight small ordered domains,

the spectrogram intensity has been normalised inside each 100 s range. These phases can also be

identified on spatially resolved optical measurements (focussed optical fibres) and show roughly

the same pattern. From Fig.4.15, the different phases are identified.

1. Three ordered phases P1, P2 and P3 from ∼ 40 s to ∼ 80 s.

2. Chaotic phase P4 from ∼ 80 s to ∼ 405 s.

3. High frequency phase P5 from ∼ 405 s to ∼ 435 s. This phase does not occur in each

experiments and is highly dependent on initial conditions.

4. Chaotic phase P6 from ∼ 435 s to ∼ 600 s. This phase tends to stabilise.

89


5. Regular oscillation phase P7 from ∼ 600 s to ∼ 680 s. This phase can have different

patterns and is related to the void instabilities (see Chap.5).

These phases are observed on both electrical and optical measurements as well as on high speed

video imaging confirming the correlation between an unstable plasma and the dust particle

growth. The described pattern of DPGI is robust except for the P5 high frequency phase which

is highly dependant on initial conditions such as gas purity. These phases are nevertheless quite

different from the instability modes observed by Goree’s group [59, 73] where a filamentary mode

(which could be related to our chaotic phase) is followed by a regular phase (great void mode).

43.25 43.3 43.35 43.4 43.45 43.5 43.55 43.6 43.65 43.7 43.75−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time (s)

Am

pli

tude

(a.u

.)

P1 P2 P3

Time (s)

Am

pli

tude

(a.u

.)

Figure 4.16: Transition between the first 3 ordered phases. Left: Current fundamental harmonicamplitude. Right: Optical signal near the plasma edge

1) DPGI begin with a succession of three ordered phases separated by clear transitions

(Fig.4.16). The phases P1 and P2 are short and are not detected in every experiments. On the

contrary, the phase P3 can last for a long period of time (few tens of seconds) and is regularly

observed. The three phases are well separated and evolve as a function of time. The transition

between the different phases can be well observed on the time series of the current amplitude. In

Fig.4.16, an example of transitions with a very short P2 phase is presented in order to illustrate

the P1-P2 transition and the P2-P3 transition. The P1 phase is characterised by well separated

current peaks. The transition from P1 to P2 corresponds to the growth of two small peaks

between the higher amplitude pattern. During the P2 phase, the small peaks continue to grow

while the higher ones decrease. Finally all peaks reach the same amplitude characterising the

P3 phase.

The different phases can also be seen on the frequency spectrum of the instabilities. In Fig.4.17,

the Fourier spectrogram and the Hilbert-Huang (HH) spectrum (see App.B) of the three first

phases of the DPGI are presented. These spectra do not correspond to the time series presented

in Fig.4.16 (In this figure t = 0 does not correspond to the plasma ignition but corresponds

to a time close to the DPGI beginning). As can be seen the P1 phase (from 0.5 s to 2.5 s in

Fig.4.17) has a mean frequency around 40 Hz but the frequency decreases slightly with time.

90


Time (s)

Fre

qu

ency

(H

z)

!

!

" # $ %

"&

#&

$&

%&

'&&

'"&

'#&

'$&

'%&

"&&

!'&&

!%&

!$&

!#&

!"&

&

P1 P2 P3

(a) Fourier spectrogram

Time (s)

Fre

qu

ency

(H

z)

!

!

" # $ %

"&

#&

$&

%&

'&&

'"&

'#&

'$&

'%&

"&&

!"(

!"&

!'(

!'&

!(

&

P1 P2 P3

(b) Hilbert-Huang spectrum

Figure 4.17: Fourier spectrogram and Hilbert-Huang spectrum during the three first phases ofDPGI. In this figure t = 0 does not correspond to the plasma ignition but corresponds to a timeclose to the DPGI beginning.

In the Fourier spectrogram (Fig.4.17(a)), the signal is composed of many harmonics. In the

Hilbert-Huang spectrum (Fig.4.17(b)), only the main frequencies of the signal are extracted (see

App.B). Thus two components are visible during the P1 phase, the main one with a frequency

around 40 Hz and another one of little intensity with a frequency around 80 Hz which was

also visible on the Fourier spectrum but could be interpreted as a harmonic of the fundamental

frequency. The transition from P1 to P2 occurs around 2.5s in Fig.4.17. In the Fourier spec-

trogram it results in a decrease of the main frequency from around 40 Hz to 30 Hz. However,

the time resolution of the Fourier spectrogram is very bad and the position of the transition can

only be approximate. In the Hilbert-Huang spectrum, it can be seen that the main frequency of

the signal rapidly decreases during the transition and as well as its intensity which becomes a

little bit weaker (from -2 to -4 on a logarithmic scale ). The transition is also visible for higher

frequency components with the decrease of the frequency of the second component from around

80 Hz to 58 Hz and a small increase in its intensity and the appearance of a third component

around 100 Hz. The transition from P2 to P3 (around 5 s in Fig.4.17) is characterised by a

strong increase of the main frequency of the signal that can be related to the transition from

two small peaks and one big peak to three equal peaks. This is visible on both the Fourier

spectrogram and the Hilbert-Huang spectrum. In the Hilbert-Huang spectrum, the intensity of

the two smallest frequencies rapidly decrease during the transition while the third component

observed in the Hilbert-Huang spectrum of the P2 phase gains in intensity but decreases in fre-

quency. In the Hilbert-Huang spectrum it also seems that there is a small intensity component

with a frequency around 45 Hz. Another advantage of the Hilbert-Huang spectrum is that that

transition are precisely identified in both the time and frequency domains.

The P3 phase is very robust (nearly always observed in our experiments) and lasts a sufficiently

long time to evolve significantly during the dust particle growth. The time evolution of its main

frequency is always the same: it decreases, reaches a minimum and then slightly increases again

until the DPGI enter in the chaotic regime. Furthermore, the mean frequency of the P3 phase

91


is linearly dependent on the phase duration (see Fig.4.18): the higher the frequency, the shorter

90

Frequency (Hz)20 30 40 50 60 70 80 90

10

20

30

40

Du

rati

on

(s)

50

60

70

80

Figure 4.18: Duration of the P3 phase as a function of its frequency

the time duration.

Optical measurements recording the plasma light (integrated over all wavelengths) at different

positions also show three ordered phases (Fig.4.16). Nevertheless, the phases P1, P2 and P3 have

different characteristics to those observed in the current amplitude measurement. The transi-

tion from P1 to P2 is characterised by an increase in the oscillation amplitudes. The P3 phase

is characterised by a strong amplitude modulation which is more prominent near the plasma

edge than in the plasma centre. There are also some discrepancies in frequency between the

optical signal and the current signal especially during the P3 phase. As can be seen in Fig.4.19,

the P1 and P2 phases are similar but when the DPGI enter the P3 phase, the current signal

frequency nearly triple while, on the optical signal, the main component frequency remains in

the continuity with that of the P2 phase. Furthermore, the P3 phase optical measurements are

slightly different from one region of the plasma to another. The Fourier transform of the signal

recorded by the fibre focussed on the plasma centre (Figs.4.19(c) and 4.19(d)) is not exactly

the same as the one recorded near the plasma edge (Figs.4.19(e) and 4.19(f)). Indeed, in the

central fibre signal, a main component with frequency around 26 Hz can be extracted as well

as a smaller amplitude component around 31Hz (Fig.4.19(d)). In the near plasma edge optical

signal the same components can be extracted but additional ones are also seen: one between

26 Hz and 31 Hz and another one around 3 Hz (Fig.4.19(f)). The last one corresponds to the

strong modulation observed in Fig.4.16.

These observations underline the fact that spatial considerations must be considered in order

to interpret the P3 phase. For this reason, high speed video imaging of the plasma glow during

the P3 phase has been performed. The camera was focussed on the plasma centre with a frame

rate of 646 fps (frames par second). In Fig.4.20, the evolution of the central line of the video (i.e.

the line passing through the discharge centre between the electrodes) is presented. The image

92


Time (s)

Fre

quency (

Hz)

0 20 40 60 800

50

100

150

200

Time (s)

Fre

quency (

Hz)

0 20 40 60 800

50

100

150

200

Time (s)

Fre

quency (

Hz)

50 60 70 800

20

40

60

80

100

120

Time (s)

Fre

quency (

Hz)

50 60 70 800

10

20

30

40

50

Time (s)

Fre

quency (

Hz)

0 20 40 60 800

50

100

150

200

Time (s)

Fre

quency (

Hz)

50 60 70 800

10

20

30

40

50

a)

b)

c)

d)

e)

f)

P1

P2

P3

P1 P1

P2

P3

P2

P3

Figure 4.19: Fourier spectrograms of (a) electrical measurements (current fundamental har-monic) and optical measurements (c) in the central fibre, (e) near the plasma edge. (b) Zoomof (a), (d) zoom of (c), (f) zoom of (e)

number is on the x-axis and the pixel number of the line is on the y-axis. Fig.4.20(a) presents

the evolution of the raw luminosity of the central line directly extracted from the video images.

As can be seen, bursts of luminosity appear regularly and their positions are changing through

time. Nevertheless, if the frequency between the maximum luminosity peaks is calculated inde-

pendently of the position, a frequency of around 96 Hz is found and corresponds to the frequency

of the current signal recorded at the same time of the video (not presented here). Consequently

a peak in the current signal is associated to the appearance of a region of enhanced luminosity in

the plasma and thus enhanced ionisation. In order to give a better view of the spatial evolution

of the glow luminosity, the time mean value of each of the line pixels has been subtracted to the

corresponding pixel value. The results are presented in Fig.4.20(b). As can be seen, when the

glow reaches its maximum luminosity in a certain region of the plasma, the other regions be-

come darker. Furthermore, the position of the maxima changes with time but follows a defined

pattern. It seems that the luminosity maxima move from one side of the reactor to the other.

The luminosity bursts reappear on the same side of the reactor every three luminosity bursts.

This explains why the frequency observed on the optical fibre signal is three times less than the

one observed in the current signal (Fig.4.19). A closer look shows that the maximal values of the

luminosity bursts are not the same for each burst while in the current signal each peak has more

or less the same amplitude. This can be explained if the luminosity burst are not in the plan

where the camera is focussed. Moreover it seems that there is a rotation of the burst of luminos-

ity with time. Indeed, when a burst reappears on a reactor side, its position has slightly changed

compared to the previous one in the same plasma region. Overall Fig.4.20(b) clearly indicates

this rotation with a frequency around 4 Hz. Consequently, the modulation observed by the op-

93


Frame number

Pix

el

nu

mb

er

200 400 600 800 1000

50

100

150

200

250

(a)

Frame number

Pix

el

nu

mb

er

!

!

"## $## %## &## '###

(#

'##

'(#

"##

"(# !'##

!(#

#

(#

'##

(b)

Figure 4.20: Evolution of the central line: (a) raw data extracts from the video (grey scale) (b)Same as (a) but the mean time values of each pixel of the line have been subtracted (in falsecolour from dark blue to dark red)

tical fibre is due to the slow rotation around the chamber axis of these three luminosity bursts.

A possible explanation for this rotation pattern can come from the PKE-Nefedov geometry. In-

deed, in the middle of the upper electrode, there is a grounded dust shaker (used to inject dust

particles into the plasma) which must modify the potential and can be the origin of the rotation.

2) After the three ordered phases, the DPGI enter a “chaotic” regime (called P4) (Figs.4.15

and 4.19 after 80 s) characterised amongst other things by a strong increase of the DPGI ampli-

tude. Then the amplitude slowly decreases during the whole duration of the phase. During the

P4 phase, structured oscillations appear in a transient manner. These structured oscillations

can be identified by some bright spots on the current and optical spectrograms. On the time

series, they appeared like bursts of order. In Fig.4.21(a), the transition between the P3 phase

and the P4 phase is clearly seen and is indicated by an arrow. A burst of order during the

chaotic regime has been encircled. A zoom of this burst of order is shown in Fig.4.21(b). In

order to understand the nature of these bursts of order, high speed video imaging has been

performed on the plasma glow. The camera was focussed on the centre of the discharge and the

frame rate was 646 fps. In Fig.4.22(a), the raw evolution of the luminosity of the central line of

the video images passing through the discharge centre is presented. As can be seen, it exhibits

luminosity maxima that vary with time and position. Each luminosity maximum probably cor-

responds to a peak in the current measurement. However, from image 2265 to image 2425, it

94


!" !# $" $#!"%"&

!"%"'

"

"%"'

"%"&

"%"(

Time (s)

a.u

.

(a)

!"#$ !"#% !"#& !"#' !"#(

!"#")

!"#""&

"

"#""&

"#")

"#")&

"#"*

"#"*&

Time (s)

a.u

.

(b)

Figure 4.21: Current signal measurements: (a) transition between P3 and P4 phases and encir-cles burst of order. (b) zoom of the encircled part of (a).

seems that the luminosity evolution follows a regular pattern. In order to further investigate

this observation, the evolution of the luminosity of the line pixels relative to their mean values is

shown in Fig.4.22(b). As can be seen, the ordered domain corresponds to the following pattern:

appearance of a high luminosity central glow, then fast decrease of the central luminosity and

appearance of two bright regions on the edge of the plasma while the centre of the discharge

is dark and finally a fast decrease of these luminous regions and the appearance of two new

luminous regions (one close to the discharge centre and the other one very decentred toward

the edge of the plasma) with a darker region in between. The glow can serve as an indicator

of the energetic electrons that cause electron-impact ionisation. Consequently, the fact that the

glow is modulated suggests that we are observing a type of ionisation wave or striation. Thus,

this three step pattern should correspond to the three peaks pattern observed during the burst

of order in the current signal due to these localised high ionisation regions. To compare our

observation with the filamentary mode reported by Goree’s group [59], a Fourier spectrum of a

4 seconds span over the chaotic regime (from 94.5 s to 98.5 s) has been performed (Fig.4.23).

In Fig.4.23(a), a noisy main frequency around 30 Hz is observed. This frequency corresponds to

the frequency of the three peak structure of the bursts of order emerging regularly during the

chaotic regime. However the spectrum extends over a wide band of frequencies. In Fig.4.23(b),

the corresponding spectrum of the central optical fibre signal exhibits the same main frequency.

However it extends over a narrower band than the current signal. It confirms the spatial aspect

of the instabilities observed clearly on high speed video imaging as the fibres are localised on a

defined region of the plasma. These Fourier spectra are very similar to the ones obtained for the

filamentary mode [59]; even so in our experiments ordered domains have been observed. Thus

we can surmise that the chaotic regime we observe could be similar to the filamentary mode.

3) In some experiments, the chaotic regime is suddenly interrupted by strong frequency and

95


Frame number

Pix

el n

um

ber

2200 2250 2300 2350 2400 2450 2500

50

100

150

200

250

Ordered domain

Image number

(a)

Frame number

Pix

el n

um

ber

!

!

""## ""$# "%## "%$# "&## "&$# "$##

$#

'##

'$#

"##

"$# !'##

#

'##

Ordered domain

Image number

(b)

Figure 4.22: Ordered domain during P4 phase: (a) raw line evolution from the video images.(b) line evolution from the video images where the mean time values of each pixel have beensubtracted

0 50 100 150 200Frequency (Hz)

Sp

ectr

al

po

wer

(a

.u.)

(a)

0 50 100 150 200Frequency (Hz)

Sp

ectr

al

po

wer

(a

.u.)

(b)

Figure 4.23: Fourier spectrum of the first chaotic regime. (a) on current signal (b) on centraloptical fibre signal

96


amplitude changes (Fig.4.15 around 405 s). This P5 phase is not always observed but when

it is present, its characteristics are nearly always the same. This new phase appears after a

continuous decrease of the P4 phase amplitude and is defined by a radical change in the DPGI

frequency and amplitude which turns into high frequency (around 500 Hz) and low amplitude

oscillations. These changes happen when no fast modifications in dust particle density and size

are expected. Furthermore, the frequency increases with time. This increase is most of the

time nearly linear but non-linear increases are sometimes observed. As already mentioned, this

phase does not always appears in the DPGI which means relatively precise conditions must be

fulfilled for its existence. Small modifications of the discharge parameters can easily turn the

high frequency phase into a chaotic one or regular oscillations.

4) After the high frequency regime a second chaotic phase (P6) is usually observed. It is

similar to P4 but exhibits more and more ordered regions.

5) The DPGI ends when the P6 phase tends to stabilise into regular oscillations. Two dif-

ferent behaviours can occur. The first and most common one is that the the transition appears

as a small and continuous increase of the DPGI main frequency. At the end of the transition,

the signal exhibits regular oscillations. They are related to are self-excited oscillations of the

dust void (region in the centre of the discharge completely free of dust particles). The second

behaviour occurs in some experiments and is slightly different. Instead of a frequency increase,

the frequency decreases and the DPGI enter a regular oscillation phase. Video imaging of the

dust cloud shows that this behaviour can correspond to large oscillations of the void or can

correspond to rotation of the void in the horizontal plane around the axis of symmetry of the

chamber. They are due to the growth of a new generation of dust particles inside the dust void.

In both cases, these instabilities have typical signatures in current and optical signals, and high

speed video imaging. They are studied in detail in Chap.5.

Regimes of the DPGI

Two regimes of DPGI have been observed: a slow regime and a fast regime. Typical frequen-

cies of the fast regime are generally 2-3 times higher than the slow ones (Fig.4.24). Nevertheless,

these regimes are not strictly separated and DPGI frequencies can spread over a wide range.

It has also been found that DPGI phase duration times are much longer for the slow regimes

than for the fast regimes. Fig.4.18 shows that the higher the frequency of the P3 phase, the

shorter its duration. The frequencies of the first phases of the DPGI can also be linked to the

appearance time of the DPGI. As the first phase P1 can be very short (less than 0.1 s), it cannot

be used to build statistics. For this reason, the appearance time of the P2 phase as a function of

its frequency (around 48 s in the left diagrams of figure 4.24 or around 60 s in the right diagrams

of figure 4.24) has been plotted in Fig.4.25. It shows that the high frequency P2 phase appears

faster than the low frequency P2 phase.

97


Time (s)

Fre

quency (

Hz)

0 20 40 60 80

50

100

150

200

Time (s)

Fre

quency (

Hz)

0 20 40 60 80

50

100

150

200

Time (s)

Fre

qu

en

cy (

Hz)

0 50 100 150

50

100

150

200

Time (s)

Fre

qu

en

cy (

Hz)

0 50 100 150

50

100

150

200

Figure 4.24: Left: Electrical signal Fourier spectrum of the current amplitude (Top) and Op-tical signal Fourier spectrum (Bottom) for fast regime DPGI. Right: Electrical signal Fourierspectrum of the current amplitude (Top) and Optical signal Fourier spectrum (Bottom) for slowregime DPGI.

30 40 5020

30

40

50

60

70

80

Frequency (Hz)

Appea

rance

tim

e (s

)

0 10 20

Figure 4.25: Growth instability appearance time as a function of P2 mean frequency

98


The behaviour of the DPGI can also be predicted by looking at the evolution of the current be-

fore the DPGI appears. Indeed, looking at the evolution of the current over first the 30 seconds

after the discharge ignition (before the DPGI start), different patterns can be seen (Fig.4.26). In

the case of a discharge exhibiting the fast regime DPGI, the current amplitude firstly decreases

during ∼ 5 s then stabilizes for ∼ 10 s and finally decreases again. In the case of a discharge

exhibiting the slow regime DPGI, the current amplitude does not decrease immediately after

plasma ignition. It starts to decrease after ∼ 1 s but very slowly compared to the fast regime

DPGI and after ∼ 5 s even starts to increase again. Finally after ∼ 15 s, the current amplitude

slowly decreases.

A possible explanation for the differences between the two regimes is a different dust particle

0 5 10 15 20 25 300.975

0.98

0.985

0.99

0.995

1

Fast regime

Time (s)

No

rma

lize

d c

urr

en

t a

mp

litu

de

(a

. u

.)

0 5 10 15 20 25 300.975

0.98

0.985

0.99

0.995

1

Time (s)

Slow regime

No

rma

lize

d c

urr

en

t a

mp

litu

de

(a

. u

.)

Figure 4.26: Left: Normalized fast regime current amplitude. Right: Normalized slow regimecurrent amplitude

growth rate and thus a variation of the dust density (nd)/dust size (rd) ratio in the plasma

from the first instants after the discharge ignition. Indeed, it has been shown that dust particle

growth is very sensitive to gas purity. Thus slight differences in gas purity from one plasma run

to another can induce important deviations in dust density and size. In the case of the slow

regime DPGI, a reduced dust growth rate can be assumed. Thus, electron attachment onto

dust particles is lesser (smaller dust particles density and/or smaller particles) and the decrease

of current amplitude is not as significant as in the case of the fast regime DPGI for which the

growth rate is supposed to be larger. Low frequencies of the DPGI phases indicate that the dust

density for the slow regime should be lower. As these frequencies are less than 100 Hz, they

should be related to the dust plasma frequency which is proportional to√

nd/rd. Thus if this

ratio is small, the dust frequency is also small. Ref.[59] proposed that the instabilities related

to dust particle growth begin when the dust particle radius rd reaches a critical value. Thus,

for reduced growth rates it takes a longer time to reach this critical dust radius. It has been

shown in previous sections that the particle growth slows down for low argon pressure. This

result was derived from the DPGI appearance time in the case of the PKE-Nefedov reactor but

was confirmed by the results of the ComPLExS reactor. This observation tends to confirm the

correlation between the dust growth rate and the DPGI regimes as slow growth kinetics have

99


already been associated with long DPGI appearance times.

Finally, our observations show that void formation is not a consequence of the DPGI as

proposed in Ref.[59]. Indeed, the void can sometimes be observed before the DPGI start. It

tends to happen when dust particle growth is very slow and in this case, the DPGI evolution

scheme does not exhibit all the phases described previously.

4.2.2 Instabilities in ComPLExS

DPGI were also observed in the ComPLExS reactor. They appear typically a few tens of

seconds after the discharge ignition. As can be seen in Fig.4.27, DPGI can be identified by

strong oscillations of the self bias voltage of the powered electrode. By looking at the plasma

glow, they can also be linked to a strong spatiotemporal modulation of the plasma luminosity

(not presented here). It is worth mentioning that DPGI were mainly observed when the pumping

time between experiments exceeded 12 hours (Regimen A) confirming the importance of the gas

purity. However, due to the the limit of resolution of the digital oscilloscope available in the

Complex Plasma Laboratory, a complete study of this DPGI was not possible.

! "! #!! #"!!!$#"

!!$#

!!$!"

!

!$!"

Time (s)

Vd

c (

x100 V

)

Figure 4.27: Evolution of the self-bias voltage in the ComPLExS reactor for P=1.2 Torr andVpp =240 V. Instabilities during the growth appear as strong oscillations in the self-bias voltage

4.3 Conclusion

In this chapter, the growth of dust particles by RF sputtering in two different RF discharges have

been investigated experimentally. It has been shown that the presence of dust particles changes

drastically the discharge properties and the plasma parameters. The dust growth kinetics can be

monitored by following the discharge characteristics (self-bias voltage, current amplitude, glow

modification (focussed optical fibre and high speed video imaging)) and the plasma parameters

deduced from Langmuir probe and spectroscopy measurements. It has been thus shown that

the growth of the dust particles modify consequently the electron energy distribution function

(EEDF). The presence of growing dust particles increases the effective electron temperature

of the plasma while the high energy tail of the distribution is thought to be depleted due to

100


charging of the dust particles.

The influence of plasma conditions have also been studied. It has been shown that high pressures

are favourable to the dust particle growth in flowing and static gas. The gas purity plays also

an essential role in the dust particle growth process. It has indeed been reported that very low

base pressure and long pumping time between experiments are necessary to achieve high dust

density and fast dust growth kinetics. The influence of the input power or RF peak to peak

voltage have also been investigated. It has been found that high power or large RF peak to peak

voltage tend to favour dust particle growth.

Finally, it has been shown that the growth of dust particles can lead to the development of

instabilities (DPGI), the evolution of which are directly linked to the dust particle growth

rate. The DPGI can be composed of a succession of ordered and disordered phases which were

observed in the different studied discharges. However, a complete description of the DPGI

evolution pattern has been done only for the PKE-Nefedov chamber due to time and equipment

restrictions.

101



Dans ce chapitre, la croissance de poudres par pulverisation dans des decharges RF capa-citives a ete etudiee experimentalement. Il est demontre que la presence de poussieres modifiegrandement les parametres du plasma et de la decharge. La croissance des poudres peut etresuivie en mesurant certains parametres. Ainsi, la tension d’autopolarisation decroıt (en valeurabsolue) durant la croissance des poudres a cause de la capture des electrons du plasma par lespoudres qui se chargent negativement. La reduction de la tension d’autopolarisation allonge laperiode de temps autorisant un flux d’electrons sur la cathode et entraıne une diminution du fluxd’ions necessaire au nouvel equilibre. L’amplitude du courant RF diminue aussi lors de la forma-tion des poussieres confirmant la diminution de la densite electronique. La collecte des poudres etleur mesure par microscopie electronique ont confirme que les poudres grossissaient avec le tempsdans le plasma mais la variation rapide de la tension d’autopolaristion n’est pas associee a uneaugmentation rapide de la taille des poussieres. Des mesures de sondes de Langmuir associees ade la spectrocopie d’emission ont mis en evidence que la fonction de distribution en energie deselectrons (FDEE) n’est pas maxwellienne et que la temperature electronique effective augmentedurant la croissance des poudres. Les mesures spectroscopiques confirment l’augmentation de latemperature electronique mais suggerent la perte des electrons energetiques de la queue de laFDEE.L’influence de differents parametres de la decharge sur la croissance des poudres a aussi eteetudiee. Ainsi, les pressions elevees sont favorables a la formation de poussieres que ce soit avecou sans flux de gaz. De plus, des pressions de base tres basses ainsi que de longues periodes depompage avant chaque experience sont necessaires pour obtenir des densites de poudres eleveeset des cinetiques de croissance rapides. On peut donc en deduire que la purete du gaz joue aussiun role essentiel dans le processus de croissance. L’influence de la puissance injectee ou de latension de polarisation RF a aussi ete etudiee. Il semblerait que de fortes puissances ou destensions RF elevees favorisent la formation des poussieres.Enfin, la croissance de poudres peut declencher des instabilites du plasma : les instabilitesde croissance des poudres. Ces instabilites suivent un schema devolution precis constitue dedifferentes phases d’amplitudes et de frequences definies. La valeur des frequences associeesaux instabilites indique que les poudres jouent un role majeur dans leur apparition et leurdeveloppement. Certaines de ces phases peuvent etre comparees aux modex filamentaire et “greatvoid” precedemment observes durant des experiences ou les poudres etaient formees par pulve-risation RF [74]. Deux regimes d’instabilites sont observes : un regime lent caracterise pardes basses frequences et une evolution lente, et un regime rapide caracterise par des hautesfrequences et une evolution rapide. Ces regimes peuvent etre relies a la cinetique de croissancedes poudres, une cinetiques lente induisant le regime lent (les frequences faibles impliquant desdensites faibles et/ou des poudres de grandes tailles).

102

Chapter 5

Dust cloud instabilities in a complex

plasma

After the growth of the dust particles, the gap between the electrodes is filled with a dense

cloud of dust particles. The discharge therefore behaves in a completely different manner than

a pristine discharge. As the dust particles are charged, the plasma can exhibit many new phe-

nomena directly linked to the dust particles. As the dust particles are massive compared to the

other species of the plasma, the phenomena involving dust particle dynamics are relatively slow

allowing experimental observation at the fundamental level.

In a complex plasma, a region free of dust particles in the centre of the discharge called a ”void”

is very often observed [59, 61, 73]. The void is characterised experimentally by a sharp boundary

where the dust density changes abruptly (Fig.5.1). Much theoretical work has been devoted to

Figure 5.1: Void region in typical dust particle clouds with various sizes of grown particles withcrystalline regions and vortex motions (Image from Ref. [61]).

the formation and behaviour of dust voids in complex plasmas [8, 58, 76, 142–147]. The current

theory can be summarised as follows: the void is thought to be triggered by an ionisation insta-

bility. The ion drag force due to ions flowing from the plasma is thought to maintain the void

by balancing the electrostatic forces on the charged dust particles, pressure gradient forces and

103

5.1 CHAPTER 5. DUST CLOUD INSTABILITIES IN A COMPLEX PLASMA

other forces on the dust particles at the sharp boundary of the void. It has also been suggested

that convective effects can have a significant influence on void formation, size and evolution

[146, 148]. However there is no unified theory for the dust void formation.

Under some conditions, the stable void can be disturbed and exhibits oscillations. This insta-

bility is named the heartbeat instability due to the contraction-expansion sequence of void size

which resembles the heartbeat motion of living bodies [58, 60, 76]. It was first reported during

microgravity experiments with injected micrometre dust particles [58] and has since been stud-

ied in laboratory experiments with grown sub-micrometer dust particles [60, 66–68]. In these

experiments, the investigations have been performed using both electrical measurements and

spatially resolved optical diagnostics and have revealed the complex behaviour in the heartbeat

instability and roughly described the dust particle motion. Furthermore, it has been found that

the self-excited heartbeat instability exhibits threshold behaviour [67, 68]: it was shown that

the instability can be triggered by increasing the power or decreasing the pressure. However, the

existence of the heartbeat instability is not fully understood theoretically and some experimental

data are still not well understood (the sharp peak in RF current signal for example).

In a discharge where the dust particles are grown by RF sputtering, continuous growth of dust

particles inside the plasma bulk can occur if the discharge parameters are suitable. Thus, even

if a dense cloud of dust particles is already present new dust particles can grow in the discharge.

In chemically active plasmas, it has been reported that the dust particles growth follows a gener-

ation scheme, i.e. dust particles are grown in a homogeneous group (small dispersion in size and

shape) called a dust particle generation which appears regularly during the life of the discharge

[37, 149]. It has been shown that a new generation of dust particles grows in the void region

of the old dust particle cloud [149, 150]. Successive generations of dust particles have also been

reported in discharges where the particles were grown by RF sputtering and the new generations

also appeared in the the void of the older dust particle cloud [61].

In this chapter, instabilities of the so-called “void” region of a dusty plasma are reported.

The next two sections are devoted to the experimental studies of two types of void instability.

• The first type of instability is a self-excited instability of the dust void (the heartbeat

instability). It was first reported in microgravity with injected dust particles (Ref. [58]

and references therein) and has been observed on ground experiments with dust particles

directly grown in the plasma [60]. Here, different regimes of the heartbeat instability are

reported.

• Sometimes the appearance of a new generation of dust particles triggers an instability of

the older dust void. Here, two classes of instability due to the growth of a new generation

of dust particle inside the void (the delivery instability and the rotating void) are reported.

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5.1 The heartbeat instability

The equilibrium between the inward electrostatic force and the outward ion drag force maintain-

ing a stable void can in some conditions be disturbed and the void size exhibits oscillations. This

self-excited instability is called the heartbeat instability and has been observed during micrograv-

ity experiments, studied in the laboratory and investigated theoretically [58, 60, 76, 151, 152].

Typical unstable voids have a size of about a few millimetres. Beyond a certain size (> 1 cm),

voids are usually stable and no instability has been observed in the parameter range explored:

their stability cannot be disturbed by decreasing the pressure or increasing the power (such

changes are known to be able to initiate the heartbeat instability [60]).

The heartbeat instability has different signatures, and measurements (electrical and optical)

show a wide range of signal shapes and frequencies. Nevertheless, two extreme cases exist:

• A high repetition rate (HRR) which consists of a continuous motion of the void and

surrounding dust particles. It occurs when the heartbeat instability is well-established

and the power delivered by the RF generator is high.

• A low repetition rate (LRR) which consists of contraction-expansion sequence (CES) of

the void size well separated in time. In between each sequence, the original conditions (i.e.

stable void) are nearly restored. It occurs when the instability tends to slow down before

stopping when the power delivered by the RF generator is low.

In between these two extreme cases (which we name “classical heartbeats”), the heartbeat

instability can exhibit failed contractions between full sequences.

The extreme cases are studied in Sec.5.1.1 and failed contractions are studied in Sec.5.1.2. The

experiments have been performed in the PKE-Nefedov chamber where the heartbeat instability

is easily observable through large windows. The heartbeat instability was characterised by its

discharge current amplitude signature and was also studied by a direct visualization of the

void and its surrounding dust cloud as well as visualization of the plasma glow in the whole

interelectrode volume.

5.1.1 Classical heartbeats

Isolated contraction (or low repetition rate (LRR))

Images extracted from a LRR heartbeat video and showing the different steps of a contraction-

expansion sequence (CES) are presented in Fig.5.2(a). The video was taken at an angle of

∼ 30 with respect to the laser sheet that was illuminating the dust cloud in the PKE-Nefedov

chamber and consequently, the aspect ratio is distorted due to the angle of view compressing

the horizontal direction. The image numbers appear on the top left hand side of each image.

The frame rate of the video was 1,789 fps.

The stable void position is determined from image 30 and an ellipse was drawn on each image in

order to be able to compare the void shape and size to its stable configuration. The CES starts

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at image 40 where a small but detectable motion of the void boundaries is observed. In image

42, the motion appears clearly and the void boundaries are now inside the drawn ellipse. The

contraction lasts up to image 55 where the void reaches its minimum size (the contraction time

is ∼ 9 ms). From this instant, the dust-free inner part slowly increases in size while a light grey

corona region (detectable from image 80) slowly reduces in size. These two regions meet and a

stable open void is regained in image 265 (the ”reopening” time is ∼ 120 ms)

In order to better depict a CES, the time evolution of a column profile is shown in Fig.5.2(b).

(a)

Image number

Pix

el num

ber

100 200 300 400 500 600

50

100

150

200

250

(b)

Figure 5.2: LRR heartbeat instability observed on the dust cloud. (a) Some images of thevoid region during a CES. The blue ellipse delineates position of the stable open void. (b) Timeevolution of the central column profile (vertical line passing through the void centre) constructedfrom the entire video (False colours from dark red to bright yellow)

This false colour image was constructed by extracting the vertical line going through the centre

of the void from each frame of the video. Time is represented by the image number on the

x-axis (1 frame is ∼ 0.56 ms) and the column profile is given on the y-axis. This representation

gives in one single image a clear characterisation of the instability in both space and time.

The low dust density region appears dark while the high dust density region are bright. Three

CES can be clearly seen and the corresponding frequency is about 7 Hz. Before the first CES

(images 1 to 40), high dust density regions are observed at the void boundaries. Then, the

CES starts: the contraction takes place and the void reaches its minimum size approximately

2.5 times smaller than the stable void region. From this moment, the size increase of the dark

dust-free inner part and the size decrease of the intermediate corona region are well resolved:

the regular size increase of the dust-free inner part can be correlated with void size increase

(i.e. a void ”reopening”) assuming that the term ”void” refers only to the zero dust density

region. From Fig.5.2, the corona region seems to correspond to the motion of dust particles

attracted to the plasma centre during the contraction and returning to their original position:

this motion is characterised by a moving boundary between a region of high dust density and

a region of low dust density. The speed of the moving boundary decreases when it approaches

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the original position. The dust particles which come closest to the centre during the contraction

react last with the smallest speed. As can be seen, the corona region at its maximum extension

is much larger than the stable void. It clearly indicates that the dust particle cloud is affected

over a long distance and not only in the vicinity of the void boundary. Collective motions can

take place over long distances due to the strong interactions between the dust particles and

their strong coupling with plasma parameter changes. Finally, Fig.5.2(b) shows that the inner

and corona regions meet (around image 280) and high dust density boundaries are restored just

before another CES occurs.

To complete these observations, the interference filter was removed from the high speed camera

(a) (b)

Figure 5.3: LRR heartbeat instability observed on the plasma glow (in false colour from darkred to bright yellow). (a) Some images of the void region during a CES (image 354 is equivalentto image 30 in Fig.5.2). The blue ellipse delineates position of the stable open void. (b)Intermediate images of (a) during fast plasma glow modifications

in order to record the plasma glow evolution without changing the camera position. The acquired

video was obtained ∼2 s after the first one. The electrical measurements were used to check

that the instability characteristics (frequency and shape) remained unchanged. In order to

synchronise the dust cloud images with the plasma glow images, the small residual plasma

glow signal passing through the interference filter was used (slightly visible in Fig.5.2(b) at the

beginning of each contraction). Typical images are presented in Fig.5.3(a). In order to bring out

changes, a reference image corresponding to a stable void configuration is subtracted from each

image and false colours are used. The plasma glow images presented in Fig.5.3(a) correspond

to the dust cloud images presented in Fig.5.2(a) (image 30 in Fig.5.2(a) corresponds to image

354 in Fig.5.3(a)and so on). The ellipse corresponding to the stable void that was used on the

dust cloud images is also superimposed onto the glow images to facilitate the comparison.

From Figs.5.2(a) and 5.3(a) it clearly appears that the void contraction corresponds to a glow

enhancement in the plasma centre. Intermediate images of the fast glow variation occurring

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during the contraction are shown in Fig.5.3(b). It appears that the glow is enhanced in a region

bigger than the original void even though the larger values are inside the void region. This

phenomenon can explain why the corona region observed in Fig.5.2 has a maximum extension

bigger than the original void since the plasma is affected over a large volume. The first image

showing a glow increase is image 361 in Fig.5.3(b). At this time, the enhanced glow region is

already bigger than the drawn ellipse. Due to camera speed limitations (0.56 ms per image) some

data are missing between images 360 and 361. Consequently, it cannot be concluded, whether

at the very beginning the glow enhancement is bounded by the void and then propagates, or

it affects directly a large volume of the plasma bigger than the void region. After reaching its

maximum luminosity (around image 364) the plasma glow decreases and reaches a minimum

below the mean glow value (darker image 369 in Fig.5.3(a)). Due to dust inertia, the void

continues to shrink (Fig.5.2(a)) but when the central glow (i.e. ionisation and consequently

ion drag force) starts to increase slowly, the contraction stops and the void starts to reopen

(image 55 in Fig.5.2(a) and image 379 in Fig.5.3(a)). The column profiles of these different

Figure 5.4: LRR heartbeat instability: current amplitude superimposed on column profile for(a) only dust cloud; (b) dust cloud and plasma glow; (c) plasma glow only. Parts (a), (b) and(c) are in false colours from dark blue to red

series are compared in Fig.5.4. This figure shows synchronised profiles obtained from videos

showing (a), the dust cloud (obtained using the laser and the interference filter), (b) the dust

cloud and the plasma glow (using the laser and the camera without the interference filter) and

(c) the plasma glow only (neither laser nor interference filter). The amplitude of the discharge

current is superimposed with its mean value (dotted line) corresponding to the stable open void.

There is a particularly good correlation between the current and the plasma glow as the main

variations are related (Fig.5.4(c)):

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• The fast current increase (∼ 0.5 ms) corresponds to the glow enhancement; it is followed

by a decrease in both sets of data.

• A small slope change appearing in the current amplitude is related to a faster decrease of

the central plasma glow.

• Both parameters values reach their minimum at almost the same time

• Finally, the plasma glow increases to its mean value and the current amplitude slowly

tends towards its mean value

The void contraction is therefore related to glow and current amplitude increase (Figs.5.4(a)

and 5.4(b)), implying that it is correlated to an increase of the ionisation in the plasma centre.

As mentioned, global variations of the glow and the current amplitude are similar but some

discrepancies exists due to the fact that the current integrates all the plasma and thus contains

information about plasma variations occurring outside the plasma centre.

Consequently, in order to see the global evolution of the plasma glow and to be able to identify

some side effects already suggested (relation between a sharp peak in the current amplitude and

glow enhancement in the horizontal direction near plasma boundaries [60]), the plasma glow has

been recorded over the whole inter-electrode space. The camera was facing the PKE-Nefedov

reactor and focussed on the centre of the chamber.

The total plasma glow recorded by fast video imaging (1,789 fps) during CES is presented

in Fig.5.5(a). The CES was not exactly the same as the one described previously but was

very similar (the frequency was lower). The presheath region above the upper and the lower

691 692 694 695 696 697

698 699 700 703 704 705

706 708 709 710 711 713

714 715 717 718 719 720

722 723 724 726 727 728

729 731 732 733 735 736

(a)

691 692 694 695 696 697

698 699 700 703 704 705

706 708 709 710 711 713

714 715 717 718 719 720

722 723 724 726 727 728

729 731 732 733 735 736

(b)

Figure 5.5: Total plasma glow during a CES of LRR heartbeat instability: (a) some raw imagesin grey scale directly extracted from the video; (b) same image as (a) with post-processing andfalse colours (from dark red to bright yellow)

electrodes can clearly be seen in the images in Fig.5.5(a). They correspond to bright regions at

the top and the bottom (the electrodes cannot be seen). Before the CES, the central plasma

region is uniform with moderate brightness (image 691). Then the glow increases quickly and

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concentrates in the discharge centre before disappearing and leaving a darker region in the centre

as described previously. In order to emphasise the changes in the plasma luminosity, the glow

of a stable configuration has been chosen as a reference image and has been subtracted from

every other images. The results are presented in false colour in Fig.5.5(b). As can be seen,

the relatively homogeneous glow (image 691) changes into a bright central region (i.e. with

higher ionisation) and dark edges (i.e. with lower ionisation) during the contraction. A reverse

situation (dark centre and bright edges) starts just before the reopening (image 699). Finally

the plasma glow slowly returns to a uniform configuration.

The time evolution of a line profile is presented in Fig.5.6(a) and magnified in Figs.5.6(b) and

Figure 5.6: LRR heartbeat instability: (a) Line profile (in false colours from dark blue to red)extracted from the image series presented in Fig.5.5; (b) and (c) are different magnifications of(a)

5.6(c). This image was constructed by extracting the line passing through the middle of the

discharge from each video frame. This line was chosen because it passes trough the region with

the biggest luminosity variations. In order to emphasise the glow variation, the temporal average

value of the pixels of the line was subtracted from each corresponding pixel. A smoothing filter

was also applied for noise reduction and a better delineation of the changing regions. Column

profiles were also extracted for the plasma glow analysis but are not presented here since no

original information could be obtained partly due to limitations imposed by the electrodes.

Nearly isolated CES often have very short time duration. In Fig.5.6, the frequency of the

instability is around 3 Hz. In between each CES, the original conditions have time to be restored.

The complete analysis of the evolution of the plasma glow is easily performed from Figs .5.6(b)

and 5.6(c). First some small oscillations are observed before the real CES and correspond to

failed contractions appearing near the instability threshold. This behaviour does not change the

following analysis and is described in details in Sec.5.1.2. Fig.5.6 confirms that when the void

contracts, the glow suddenly concentrates leaving dark edges. The void reopening occurs during

the reverse situation with two symmetrical bright regions surrounding the plasma centre. Then

these regions diffuse slowly toward the centre until a homogeneous glow is obtained once again.

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The correlation with the current amplitude is shown if Figs.5.7(a) and 5.7(b) with non-smoothed

Figure 5.7: LRR heartbeat instability: (a) Line profile (in false colours from dark blue to red)calculated from the image series presented in Fig.5.5 with the current amplitude superimposed;(b) zoom of a; (c) current amplitude; (d) central line (line 178) profile of (a)); (e) middle positionline (line 105) profile of (a)

images in order to get pixel precision. The dotted line represents the direct component of the

current amplitude. As can be seen, the void contraction corresponds to an increase of the

current and thus an increase of the global ionisation. Indeed an increase of the central plasma

glow corresponds to an increase of the current amplitude meaning that the central increase is

not compensated by the glow decrease at the plasma edge. Then, when the reverse situation

occurs, the ionisation appears to be below the stable situation value (current amplitude is below

its mean value): the dark central region is not balanced by the larger but slightly brighter

near edge regions. Finally these regions tend to rejoin in the centre following the slow increase

in the current amplitude. These behaviours are also visible by comparing the evolution of

the current amplitude with line profiles (Figs.5.7(c)-(e)) extracted from Fig.5.7(a) (not to be

confused with the temporal evolution of the line profile extracted from the video images) and

physically corresponding to the time evolution of individual pixels taken in (d) the plasma centre

and (e) the near edge.

Successive contractions (or high repetition rate (HRR))

The analysis of a HHR heartbeat sequence is more difficult as the dust cloud is moving fast and

constantly. Consequently, the dust cloud has no clear stable position. In Fig.5.8, images and

column profile (i.e. vertical line passing through the void) have been extract from a high speed

video of a HRR heartbeat. The frame rate was 1,789 fps. In order to increase the contrast, false

colours have been used to plot the extracted images. The frequency of the instability was in

this case 31 Hz. As can be seen, it clearly appears that the dust cloud has a constant motion.

Indeed, horizontal bright regions , corresponding to high dust density void boundaries, are not

observed in Fig.5.8. Furthermore, an inner part of the void region which is always free of dust

can be identified. This region exhibits only small variations of its size. Dust motion takes place

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69 71 72 74 76 78

79 81 83 84 86 88

89 91 93 95 96 98

100 101 103 105 106 108

110 112 113 115 117 118

120 122 123 125 127 129

(a)

Image number

Pix

el num

ber

20 40 60 80 100 120 140

50

100

150

200

250

(b)

Figure 5.8: HRR heartbeat instability observed on the dust cloud: (a) some images of the voidregion (in false colours from dark red to bright yellow; (b) time evolution of the central columnprofile constructed from the entire video.

in a low dust density region between the dense dust cloud and the dust-free inner part of the

void. In the LRR heartbeat sequence, this region has been identified as dust particles going back

to their original position. In the HRR heartbeat, a new contraction of the void occurs while

the dust particles that entered the void during the previous contraction are still inside and try

to reach their original position. As can be seen in Fig.5.8(a), a new contraction starts around

image 69 (complex motion does not allow precise determination) while dust particles are still

inside the void region. At this time the plasma glow is close to its maximum value in the centre

(determined from the video of the plasma glow correlated to the present one). This complex

behaviour leads to the relatively constant regions observed in Fig.5.8(b): a dust free black inner

region and a low dust density grey intermediate region as the constant motion of the dust cloud

maintains more or less constant spatial dust density.

The analysis of the entire plasma glow between the electrodes was also performed for a HRR

heartbeat. The experimental procedure was the same as the one described previously for a LRR

heartbeat. The frame rate was 1,789 fps. The images extracted from the video are presented in

Figs.5.9(a). Fig.5.9(b) was constructed with the same post-processing procedure as Fig.5.5(b)

(subtraction of a reference image).

In a HRR heartbeat, the frequency of the CES is such that no exact return to the original stable

conditions is achieved and the next CES occurs while the void has not reached its stable state.

However the sequences last longer (i.e. there are more CES cycles but the different parts of each

CES are not as abrupt as in a LRR heartbeat where the sequence are almost isolated cycles),

a better time resolution is obtained and the variation of the plasma glow is well in evidence

(Figs.5.9(a) and 5.9(b)). Indeed, for example, the central glow increase lasts for 3 images (∼ 1.5

ms) in the LRR case while it lasts for 17 images (∼ 9 ms) in the HRR case. In comparison with

the LRR case, the change from bright centre and dark edges to the reverse situation is more

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160 163 166 169 171 174

177 180 183 186 189 191

194 197 200 203 206 208

211 214 217 220 223 226

228 231 234 237 240 243

246 248 251 254 257 260

(a)

160 163 166 169 171 174

177 180 183 186 189 191

194 197 200 203 206 208

211 214 217 220 223 226

228 231 234 237 240 243

246 248 251 254 257 260

(b)

Figure 5.9: HRR heartbeat instability observed on the total plasma glow: (a) some raw imagesdirectly extracted from the video during a CES; (b) same images as (a) with post processingand false colour (from dark red to bright yellow)

strongly marked. The different phases of the CES are the same as previously described for the

LRR case but are more distinguishable in the present case.

The extracted line profile is very significant and summarises nearly all results concerning the

plasma glow evolution during the heartbeat instability (Fig.5.10). Starting from a void expan-

sion (dark inner part and bright plasma edges), the glow slowly concentrates from the plasma

edge towards the centre. This motion can be correlated to the grey corona region observed in

the dust cloud and described previously. Furthermore, it clearly appears that the brighter edge

regions do not meet completely and no homogeneous plasma is restored as can be seen for a LRR

heartbeat (Fig.5.6). The central glow increases at the expense of the plasma edges before the

Figure 5.10: (a) Line profile (in false colour from dark blue to red) calculated from the imageseries presented in Fig.5.9. (b) Zoom of (a) with electrical measurement superimposed.

brighter glow regions have fully converged toward the discharge centre. This central enhance-

ment is then followed by a strong and fast reversal. The amplitude of the discharge current

during the instability is superimposed on the line profile in Fig.5.10(b). The bright regions mov-

ing from the plasma edge towards the plasma centre correspond to a continuous increase of the

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current. Then it appears that the strong reversal of the brightness is related to the presence of a

sharp peak in the current. Measurements without this peak certainly correspond to cases where

the brightness reversal between the plasma centre and the edge exists but is not as strongly

marked (see Fig.5.7 for example). This sharp peak is an interesting feature already observed

but not fully explained. In Ref.[60], Mikikian preliminary results suggested that this peak could

be correlated with a strong glow decrease in the centre and an increase at the plasma edge. The

measurements were performed thanks to a spatially resolved analysis of the argon line emission

at 750.39 nm (four different positions). Further results with five optical fibres (also used for

the characterisation of the dust particle growth instability (Sec.4.2) and integrating all plasma

wavelengths gave similar results [68] but no complete description was available. The present

results greatly enhance spatiotemporal resolution of this phenomenon and clearly correlate the

sharp peak with an enhancement of the plasma glow outside the central region. Variations of

the current amplitude are representative of global change in the plasma. Consequently, from

the present analysis, it appears that the discharge current is not dominated by the plasma core

only.

These effect are also seen in Fig.5.11 by direct comparison of (a) the evolution of the current am-

a.u

.a.

u.

a.u

.

0.07 0.08 0.09 0.1 0.11 0.12 0.13

Time (s)

a.u

.

Electrical measurement

Middle position line profile

Integrated central luminosity

(a)

(b)

(c)

(d)

Central line profile

Figure 5.11: Extracted data from Fig.5.10: (a) amplitude of the current; (b) central line profile;(c) line profile in the plasma edge (glow reversal region); (d) Plasma glow integrated over a smallcentral volume from complete image series of Fig.5.9

plitude with data extracted from Fig.5.10: (b) central line profile (i.e. physically corresponding

to the evolution of the signal of the central pixel), (c) middle position line profile (i.e. variation

of the signal of a pixel in the glow reversal region) and directly from the video: (d) integrated

central luminosity (i.e variation integrated over a small centred area assumed to represent the

114


core of the plasma). As can be seen, the main plasma centre changes are observed in the current

amplitude evolution. Nevertheless, it appears that the maximum value of the central glow occurs

while the current has already started to decrease. In fact, at this time, electrical measurements

are affected by the edge plasma glow decrease as observed in Fig.5.11(c). Finally, the sharp peak

in current amplitude is directly related to the strong and fast enhancement of the plasma edges

while the centre become the less luminous part. This result confirms the preliminary results

obtained by Mikikian and Boufendi (Fig. 7 of Ref.[60]).

In order to have a good temporal resolution of the glow reversal, very high speed video imaging

(10,000 fps) of the plasma luminosity has been performed. The evolution of the luminosity of

the central line (line going through the plasma centre) is presented in Fig.5.12. In order to

highlight the glow variation, the temporal average value of each pixels of the line was subtracted

from the corresponding pixel. As can be seen in Fig.5.12, the glow reversal does not seem to

Image number

Lin

e p

rofi

le (

pix

els)

200 400 600 800 1000 1200 1400 1600 1800 2000

50

100

150

200

250

(a)

Image number

Lin

e p

rofi

le (

pix

els)

400 450 500 550 600 650 700 750 800 850 900

50

100

150

200

250

(b)

Figure 5.12: (a) Evolution of the central line luminosity of the images extract from very highspeed video (10,000 fps) during a HRR heartbeat (in false colour from blue to red). (b) zoomof (a)

correspond to a fast propagation from the centre to the edge of the plasma. Indeed the plasma

edge luminosity increases at the same time the plasma centre luminosity decreases while, in the

intermediate region, the luminosity does not vary significantly. Following the hypothesis that a

strong glow luminosity corresponds to a high ionisation rate, it indicates that the ionisation is

favoured at the plasma edges but no ionisation waves seems to propagate from the centre toward

the edge.

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Asymmetry in the void contraction during the heartbeat instability

An interesting phenomenon concerning the dust cloud motion during the heartbeat instability is

the observable delay appearing in the response of the horizontal and vertical directions. Indeed

in some conditions (currently not clearly defined) the dust cloud shrinks horizontally before

shrinking vertically [67, 68]. As an example, a record containing both the dust cloud and the

plasma glow when a stable open void suddenly became unstable and started size oscillations is

shown in Fig.5.13. The frame rate of the video was 1,789 fps and the two first CES are shown.

In Fig.5.13, image 135 clearly shows the stable open void with the corresponding drawn ellipse.

135 138 142 143 144 145

146 147 148 149 150 152

155 160 170 180 200 220

230 235 240 241 242 243

244 245 246 247 249 251

254 269 279 299 310 329

Figure 5.13: Simultaneous record of the plasma glow and the dust cloud in the void regionduring the setting up of the heartbeat instability. From image 135: stable open void (delineatedby a red ellipse) followed by a first contraction. A new contraction starts at image 241 beforethe original conditions have been restored

In image 142 the plasma glow starts increasing and reaches its maximum value at image 144.

In this last image, it appears that the plasma glow enhancement affects a region larger than

the void size and a correlation with the corona region previously mentioned can be suggested

again. This hypothesis is correct assuming that the intensity increase in the surrounding dust

cloud is entirely due to plasma glow changes and not due to any dust density changes. The

drawn ellipse allows us to see that the void first collapses in the horizontal direction (image 143)

followed by a contraction in the vertical direction (image 147). As the images are compressed

in the horizontal direction due to the angle of view of the camera, a detectable motion on

the horizontal axis clearly indicates a motion of the dust cloud and cannot be an artefact. The

plasma glow reaches its minimum around image 148 and, as already observed, the void continues

to shrink to its minimum size around image 155. From this point, the inner part of the void

slowly increases in size and the grey intermediate region, consisting of dust particles going back

to their original position, becomes gradually visible. The next CES occurs before the initial

116


conditions have been completely restored (image 241).

These various steps are well observed on the column profile shown in Fig.5.14(a). For a better

Figure 5.14: (a) Column profile (in false colour from dark red to bright yellow) extracted fromimage series presented in Fig.5.13 with central plasma glow evolution superimposed (blue curvecorresponding to line 95 of Fig.5.14(a)) (b) zoom of (a), (c) corresponding line profile.

understanding, the temporal evolution of the glow (central pixel of the void) is superimposed

at the bottom of the image. The stable void is accurately defined with its constant size and its

high dust density boundaries (between image 100 and 142). When the plasma glow increases,

the void shrinks. Its minimum size is reached while the plasma glow in the plasma centre has

already reached its minimum value and has started to slowly increase again. Then the glow stays

relatively constant in the void centre while dust particles are expelled to their original position.

It clearly appears that the void did not reach its original size when a new glow increase occurs.

This effect is characterised first by a dark inner region which does not rejoin the position of

the stable void and second by the intermediate corona region which is bigger than the original

void size. This last point means that the “wave front” formed by the returning dust particles

is still moving towards the original void boundaries. The speed of this wave front slows down

when approaching the original conditions. Furthermore, it is clear that the instability is in

a setting up phase: starting from a stable open void, the characteristics of the first and the

second CES are slightly different in terms of minimum void size and reopening time duration.

Concerning the time delay appearing in the vertical direction (see Fig.5.13) it can also be seen

by comparing column and line profiles extracted from the video. These two profiles are shown

respectively in Figs.5.14(b) and 5.14(c). Mikikian et al. suggested that the different collapse

times can be correlated with the different slopes in the current signal [67]. The reason for this

asymmetry is still unclear due to the lack of statistics and the spatiotemporal limitations of the

video acquisition. Nevertheless, one possible hypothesis is the geometry of the PKE-Nefedov

reactor and the different confining conditions in the horizontal and vertical directions. Indeed

the electrode are ∼ 4 cm in diameter and glass boundaries are ∼ 3 cm away from the electrode

edges. Consequently, the plasma usually diffuses beyond the electrodes towards the lateral sides.

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The plasma width is then nearly twice as large as the height and it is nearly the same for the

trapped dust cloud and the void. No strong electrostatic barrier exists due to the distance

to the glass boundaries and the plasma and the void changes in the horizontal direction are

relatively unconstrained. On the contrary, the electrodes impose a strong electrostatic barrier

in the vertical direction and thus changes and motion in this direction are drastically controlled

by the sheaths.

5.1.2 Failed or aborted heartbeat

It has been observed previously that sometimes the CES during an heartbeat instability are not

complete and terminate before a void contraction has occurred [68, 75]. It results in failed or

aborted peaks easily observable on the discharge current amplitude.

In Fig.5.15, the AC component of the amplitude of the discharge current has been recorded

for no failed peak in a HRR heartbeat, 1 failed peak, 2 failed peaks and 3 failed peaks. The

presented signals come from the same experiment, where the heartbeat instability exhibits more

and more failed contractions. On the time axis the 0 has been chosen at the minimum value of

the current amplitude of a complete CES in order to be able to compare its evolution during

a full CES for the different cases. As can be seen, the evolution of the current just after the

!!"!# ! !"!# !"$ !"$#!%

!&

!

&

%

'()$!

!*

Time (s)

AC

cu

rren

t (a

. u

.)

+,)-.,/012)31-4

$)-.,/012)31-4

&)-.,/012)31-45

*)-.,/012)31-45

Figure 5.15: Current during the heartbeat instability for different number of aborted peaks

sharp current decrease corresponding to the void contraction is very similar in all cases: the

current increases until it is a little above its mean value (0 in Fig.5.15 as it does not show the

DC component of the current). Then, as the signal corresponding to a heartbeat without failed

peak continues to increase and follow the same behaviour as describes previously for a HRR

heartbeat, the other signals decrease. All signals with at least one failed peak reach a minimum

at the same instant and the current starts to increase again. The signal with only 1 failed peak

exhibits then a complete sequence while the two other ones decrease again. The same difference

118


is then observed between a signal with two failed peaks and 3 failed peaks. It has to be noted

that a failed peak occurs exactly at the instant a full CES is expected. This behaviour tends

to confirm that the heartbeat instability is a threshold phenomenon. Another feature that can

be observed in Fig.5.15 is the increase of the sharpness of the major peak when the number of

failed peaks increases. Indeed, as can be seen, the duration of the fast increase followed by the

fast decrease in the current amplitude is shorter when the number of failed peaks is larger.

The transition from one aborted peak to two aborted peaks on the current signal is analysed

in details in Fig.5.16. In Fig.5.16(a), the evolution of the amplitude of the current (blue curve)

is shown. The instantaneous energy (red curve) calculated from Hilbert-Huang analysis (see

App.B) is superimposed on the current. In Fig.5.16(b) the Fourier spectrogram of the current is

plotted and in Fig.5.16(c) the Hilbert-Huang spectrum is shown. As can be seen, the transition

occurs at time t ∼ 1.5 s. In the Fourier spectrogram (Fig.5.16(b)), the transition results in a

decrease of the main frequency from 12 Hz to 8.5 Hz and the appearance of more harmonics

in the signal. In the Hilbert-Huang spectrum (HHS) (Fig.5.16(b)), the transition is also ac-

companied by a decrease of the main frequency from 11.5 Hz to 8 Hz. It worth mentioning

that in the Hilbert-Huang analysis, the intensity of a frequency at a time instant represents the

probability of a wave with such frequency to develop (see App.B). The main advantage of the

HHS is that the transition can be precisely identified in the frequency domain as it records the

instantaneous frequency (see App.B). Thus, it can be seen that the red line around 11.5 Hz at

t = 1 s decreases at t = 1.5 s which is exactly when the transition occurs on the current signal

(Fig.5.16(a)). The other advantage of the HHS spectrum is that it permits identification of the

main frequency components of the signal without their harmonics. Thus, before the transition,

the signal exhibits two main components (around 12 Hz and around 22 Hz) and after the tran-

sition it exhibits three main components (8 Hz, 16 Hz and 23.5 Hz) which are visible on both

spectra but accompanied by a lot of harmonics in the Fourier spectrum. Finally, the HHS allows

us to see that the main frequency seems to be slightly modulated. The instant energy carried by

the signal (i.e. the sum over all frequencies of the intensity of each frequency) can be extracted

from the HHS (Fig.5.16(a)). It can be seen that the main peak of the heartbeat carries the most

energy: a closer look to the HHS shows that, during the main peak, the probability of local high

frequency waves appearing is high (temporally localised red lines in regions above 50 Hz in the

HHS particularly visible around t = 2 s and below the arrows in Fig.5.16(d)). It seems that for

two aborted peaks the instantaneous energy is more important than for one aborted peak. This

two last observation can be understood by the fact that the main peak is characterised by sharp

increase and decrease of the current and in the case of two aborted peaks, the main peak is even

sharper. This sharpness is then accompanied by the high probability of existence of local high

frequency components in the signal.

The evolution of the total plasma glow between the electrodes has been recorded. The

frame rate was 894 fps. Some images extracted from the video are presented in Fig.5.17.

Fig.5.17(a) shows raw images of the plasma glow during a heartbeat sequence with two failed

119


! !"# $ $"# % %"# & &"# '!$

!!"#

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Time (s)

a.u

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()**+,-

.,/-0,-0,+1)/2+,+*34

(a) Current and instant energy

Time (s)

Fre

qu

ency

(H

z)

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!

" # $

#%

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"#%

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(b) Fourier spectrogram (the intensity varies from darkblue to red (logarithmic scale))

Time(s)

Fre

qu

ency

(H

z)

!"#$%&'!!()*+,-.%/'&(0

,

,

! " # $ %

#!

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(c) Hilbert-Huang spectrum (the intensity varies fromdark blue to red (logarithmic scale))

Time(s)

Fre

qu

ency

(H

z)

!"#$%&'!!()*+,-.%/'&(0

1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.40

50

100

150

(d) zoom of c). The green arrows indicate the position of a main peak in the current signal

Figure 5.16: Transition from one aborted peak to two aborted peaks

120


peaks. Fig.5.17(b) shows the same images with the image of the plasma glow before the first

failed peak chosen as a reference image and subtracted from every other images.

In Fig.5.17(a) the bright presheath region of the plasma are clearly visible at the top and the

150 155 158 160

165 170 175 178

180 185 190 193

195 200 205 210

(a) Raw images extract from the video

150 155 158 160

165 170 175 178

180 185 190 193

195 200 205 210

(b) Images from the video after postprocessing and falsecolour (dark red to bright yellow)

Image number

Pix

el num

ber

150 200 250

100

200

300

400

500

600

(c) line profile extract from the image series of (a). The blue dash line representthe evolution of the current amplitude.

Figure 5.17: Evolution of glow during a heartbeat with two failed peaks

bottom of the images. However as can be seen, the change in the central glow is barely visible

and only the glow increase due to the main CES is clearly identified (image 193). In Fig.5.17(b),

the relative changes are easily identified. It can be seen that for the first failed contraction, the

central glow tends to increase (image 155 and 158) while the plasma edges are less luminous.

Then the plasma edges become brighter while the central glow decreases (images 160 -165). The

main difference with the complete CES of HRR and LRR heartbeat is that, in the present case,

121


the luminosity changes both at the plasma centre and the plasma edges are smaller. Then a new

failed contraction with the same glow pattern occurs (images 170-185). However, the maximum

value of the central luminosity has slightly increased. From image 190, the main CES occurs:

a strong increase of the central glow while the edges become darker is followed by an abrupt

change to bright edges and dark centre (image 195). The glow then returns slowly to its original

configuration.

This pattern can be clearly identified on the line profile presented in Fig.5.17(c). This figure

shows the evolution of the line passing trough the plasma centre (y-axis) as a function of time

(x-axis). The time average value of each pixel in the line has been subtracted from the corre-

sponding pixel in order to emphasise the relative changes of the glow. Two main contractions

are visible in this figure. As can be seen, after a main CES, the glow converges towards the

centre (but it cannot reach the same high value as for a main contraction) before the situation

is reversed and the edges are brighter than the centre. It is worth noting that the luminosity of

plasma centre does not decrease as much as for the case of a main CES. These observations can

also be correlated with the evolution of the current amplitude. As can be seen in Fig.5.17(c),

the current increase during failed peaks is quite low and the current decrease is not very abrupt.

This may explain why the plasma centre luminosity changes are smaller. On the other hand

during a main peak the current increase is much more important and the decrease is very sharp

and the current goes well below its mean value as was already observed for LRR and HRR

heartbeats.

The motion of the dust cloud has been recorded for heartbeat sequences with two failed peaks.

The frame rate was 1,789 fps and the camera was at an angle ∼ 30 with respect to the laser

sheet. An interference filter was used in order to cut out most of the plasma light. Raw frames

from the video are not presented as they do not bring forward much information compared to

those which were presented for the HRR heartbeat. However, the evolution of the central col-

umn profile (i.e. the column passing through the centre of the void) is presented in Fig.5.18(a)

as it allows small changes in the dust cloud configuration to be seen. In order to emphasise

those changes, the time average value of each pixel in a line has been subtracted from the cor-

responding pixel (Fig.5.18(b)). The evolution of the current amplitude has been superimposed

on both figures.

As can be seen in Fig.5.18, the dust cloud is in constant motion as for a HRR heartbeat where

only a central region (between pixels 130 and 140) is always dust free (the darker region in

Fig.5.18(a)). An intermediate region filled with moving dust particles is observed between each

main contraction. As observed previously, the void starts to contract when the current and the

central glow increase strongly. The central glow increase can be seen clearly in Fig.5.18(b) where

the changes in the residual light passing through the interference filter are magnified. Never-

theless, the most interesting feature in Fig.5.18 is the behaviour of the corona region. Indeed

as can be seen, it does not exhibit the same smooth movement as observed for the LRR and

HRR heartbeats. Indeed, the failed peaks observed on the current amplitude signal correspond

to small contractions of the dust cloud which are accompanied by sudden acceleration of the

122


Image number

Pix

el n

um

ber

50 100 150 200 250 300

50

100

150

200

(a) Line profile extracted from dust cloud video (infalse colour from dark red to bright yellow)

Image number

Pix

el n

um

ber

50 100 150 200 250 300

50

100

150

200

(b) Same as (a) except that the time mean value ofeach pixels have been subtracted (in false colour fromdark red to bright yellow)

Figure 5.18: Evolution of the central column extracted from the dust cloud video during aheartbeat instability with two failed peaks. The dashed blue line superimposed to the imagesrepresents the evolution of the current amplitude.

corona region towards the centre. This is particularly clear in Fig.5.18(b).

It has been recently shown that the evolution pattern of the transition followed a mixed-mode

oscillation (MMO) scheme (alternation of small amplitude oscillations between two large ones)

and that the frequency of the heartbeat slightly decreases during a sequence with n failed peaks

[153]. In this article, it is shown that when the heartbeat instability is stopping, the number

of failed oscillations increases steadily and the progressive occurrence of the failed contractions

follows an incomplete devil’s staircase scheme.

5.1.3 Physical interpretation of the heartbeat instability

From the different analyses reported in the previous sections, several features of the heartbeat

instability can be brought to the fore.

• The void contraction corresponds to an increase of the central plasma glow. It suggests

that a sudden enhanced ionisation in the plasma centre changes the ratio between the

inward electrostatic force and the outward ion drag force: this enhanced ionisation may

increase the positive space charge inside the void region due the higher mobility of the

electrons that diffuse away much faster. Consequently the electric field in the plasma is

enhanced as is the ion flux. However, it has been theoretically shown that the ratio of

the ion drag-to-electric forces decreases rapidly with the electric field [104]. Hence, the

electrostatic inward force become predominant and the dust particles enter the void region.

The void centre becomes darker but the dust particles continue to move inward due to

their inertia with a more or less constant speed and stop suddenly (see for example 5.2).

The void is however not completely filled indicating that a force is preventing a further

123


motion towards the centre. As the gravitational force is acting over all dust particles and,

among others, those constituting the upper part of the dust cloud, the force preventing

the void closure has to be stronger than the gravitational force. A that time, the central

glow is weak and nearly homogeneous, a property usually associated with weak electric

field. In Ref.[104], it is shown that for weak electric field the ion drag force exceeds the

electrostatic force and it is reasonable to assume that the force stopping the dust particles

is the ion drag force.

• The dust particles situated near the plasma centre are then nearly immobile while a dust

particle “wave front” coming from the plasma edge is detected. At the same time, a corona

region of brighter plasma glow and thus higher ionisation is observed at the plasma edges.

This region surrounds the plasma centre and moves slowly towards it. The correlation with

the dust cloud motion indicates that this region may correspond to the corona region. An

enhanced ionisation at the plasma edges can create an outward electrostatic force in the

vicinity of the corona region if it is assumed that in analogy with the contraction phase, a

plasma region with an enhanced glow attracts the dust particles. Thus, the central dust

particles are immobile (far from the interface) while the particles close to the corona region

are attracted to it. As the dust particles move, the local conditions change and the corona

region moves inwards attracting the inner dust particles one after the other. While the

plasma homogeneity is restored, the wave front speed decreases and the last particles to

move outward are the closest to the void centre. In the LRR case, a stable void is restored

between each CES which can be considered as isolated and independent while the HRR

case leads to a continuous motion of the dust cloud as the central dust particles are unable

to reach their initial positions.

• Asymmetry of the void contraction has been observed under some conditions. This asym-

metry can find its origin in the discharge geometry. Indeed, the plasma is confined by the

electrodes in the vertical direction and can almost freely diffuse in the horizontal direction.

The corona region confirmed this hypothesis when observed on the plasma glow images:

in the vertical direction it is merged with the bright presheath region.

• The sharp peak in the current signal can be associated with the glow reversal from bright

centre and dark edges to bright edges and dark centre when this phenomena is strongly

marked. It is replaced by a change in the slope in weaker cases. This reversal is very fast

but imaging at very high speed shows that the edge glow increase starts at the same time

as the centre glow decreases. The hypothesis of the propagation of an ionisation wave

seems thus very unlikely. It is more reasonable to assume that ionisation is favoured at

the plasma edge when the dust particles enter the void region.

Failed or aborted heartbeats have been identified from the discharge current, the plasma glow

and the dust cloud motion. It has been shown that the appearance of failed contraction reduce

the frequency of the main peaks and that the sharpness of the main peaks is increased when

124


the number of failed peaks increases. The transition from n to n + 1 failed peaks is sharp in

both temporal and frequency domains. However, it has been recently shown that the evolution

pattern of the transition followed a mixed-mode oscillation (MMO) scheme (alternation of small

amplitude oscillations between two large one) and that the frequency of the heartbeat slightly

decreases during a phase with n failed peaks [153].

It has been discussed previously that for a LRR heartbeat the CES is very short. In Fig.5.6,

some oscillations are observed in the central glow of a LRR heartbeat before the main CES. We

have also mentioned that LRR heartbeats are observed when the instability is about to stop.

Consequently, when the heartbeat instability slows down before it stops, it may be possible that

a heartbeat instability evolves form HRR to LRR with a continuous increase of the number of

failed peaks following a mixed-mode oscillation scheme. Our hypothesis is, since the heartbeat

instability is a threshold phenomenon [67, 68], energy needs to be accumulated by the system and

is relaxed during a CES. Then, two possibilities can explain the stopping of the instability in a

more or less similar way: energy from the plasma is dissipated and not fully balanced by the input

power or, the plasma parameters are evolving and the instability threshold changes. In both

cases, it may be more difficult for the plasma to accumulate enough energy to trigger a full CES

and a failed peak occurs instead. It has been shown theoretically that the heartbeat instability

involves many parameters (ionisation level, ion flux, ion density, electron density, electric fields,

dust particle charge, etc) [76]. All these parameters have different time scales. It has been

demonstrated that MMO occurs in systems with multiple time scales and that varying a control

parameter (not defined yet for the heartbeat instability) can change the MMO pattern (i.e. the

number of small oscillations; see Ref.[154] and references therein). Theoretical investigations

are thus needed to fully understand the physical processes involved in the heartbeat instability

and its evolution.

5.2 Instabilities related to the growth of a new generation of

dust particles

In a discharge where the dust particles are grown by RF sputtering, continuous growth of dust

particles inside the plasma bulk can occur if the discharge parameters are suitable. Thus it

has been reported that new generations of dust particles grow in the void region in sputtering

discharges [61] or reactive plasmas [149, 150]. Under some conditions, the new generation can

trigger an instability of the old dust void.

5.2.1 The delivery instability

After dust particle growth in the PKE-Nefedov chamber, the dust particle cloud is often in the

following state (Fig. 5.19): a void in the centre of the chamber is surrounded by a cloud of dust

particles. The dust particle cloud can be composed of different dust particle generations which

125


Figure 5.19: Dust cloud used for the study of the “delivery” instability.

are clearly separated. When the discharge parameters are suitable to dust particle growth, new

generations of dust particle can regularly grow and appear in the void region. Under some

conditions (not defined yet), the growth of a new generation of dust particle can trigger an

instability of the void region: the “delivery” instability.

The “delivery” instability starts with a sudden expansion of the void size followed by a con-

traction to its original size. An expansion-contraction sequence (ECS) of the delivery instability

is shown in Fig.5.20. It presents images extracted from high speed video imaging of the dust

cloud during a delivery instability. The camera was placed at an angle of ∼ 30 with respect to

the laser sheet that was illuminating the dust cloud in the PKE-Nefedov chamber and the frame

rate was 500 fps. One of the main differences with the heartbeat instability is that, during a

delivery instability, the minimum size of the void corresponds to its size before the instability

starts while during the heartbeat instability the maximum size of the void is its stable config-

uration. As can be seen, the expansion in the vertical direction is more pronounced than in

the horizontal direction. Fig.5.21 presents the time evolution of the video image central column

going through the void extracted from the same video images presented in Fig.5.20. Time is

represented by the frame number in the x-axis (1 frame is 2 ms) and the column profile is given

in the y-axis. It is clear that the amplitude of the void oscillation increases with time during the

delivery instability. It is also worth noting that the expansion time is longer than the contraction

time. This is also the case during the heartbeat instability but the characteristic times of the

heartbeat instability are much shorter than those of the delivery instability (see Ref.[151] and

Sec.5.1) . For example, a full ECS of the delivery instability lasts for ∼ 300 ms while a full

CES of the heartbeat instability lasts for at most a few tens of milliseconds. As can be seen in

Fig.5.22, both frequency and amplitude of the instability evolve with time. These changes must

be linked to the evolution of the plasma (and/or dust) parameters during the instability. The

delivery instability ends when the new generation of dust particles reaches critical dimensions

(size and/or density of dust particles). Indeed, soon after the void stops oscillating, the new

cloud becomes visible to the naked eye. A new delivery instability can eventually start again

when a new generation of dust begins to grow. Consequently, the evolution of the plasma pa-

126


1990 1994 1999 2003 2007 2011

2016 2020 2024 2029 2033 2037

2041 2046 2050 2054 2058 2063

2067 2071 2076 2080 2084 2088

2093 2097 2101 2106 2110 2114

2118 2123 2127 2131 2136 2140

Figure 5.20: Evolution of the void during one expansion-contraction sequence of the deliveryinstability. The blue ellipse delineates position of the stable open void.

Frame number

Pix

elnum

ber

500 1000 1500 2000 2500

20

40

60

80

100

120

a)

Frame number

Pix

elnum

ber

1350 1400 1450 1500 1550

40

60

80

100

b)

Figure 5.21: a) Evolution of the central column going through the void (frame rate: 500 fps).b) Zoom of a) with enhance contrast. The plasma glow luminosity was not totally stopped bythe interference filter allow us to visualize major glow luminosity fluctuation at the same timeas the dust cloud.

127


2 2 3 3 4 40

5

10

15

Time (s)

Fre

qu

ency

(H

z)

(a)

2 2.5 3 3.5 4 4.50

5

10

15

20

25

30

Time (s)

Am

pli

tud

e (p

ixel

s)

(b)

Figure 5.22: (a) Evolution of the delivery instability frequency. The dashed curves representthe theoretical fit. (b) Evolution of the delivery instability amplitude (1 pixel ∼ 100 µm). Thedashed curves represent the theoretical fit.

rameters during the instability have to be correlated with the growth kinetic of the new dust

particle generation.

2 15 28 42 56 70

83 97 111 124 138 152

165 179 193 207 220 234

248 261 275 289 302 316

330 344 357 371 385 398

412 426 439 453 467 481

Figure 5.23: Evolution of the glow during one expansion-contraction sequence of a “delivery”instability (in false colour from blue to red). In order to improve the visibility of the luminosityvariations, the image 1 is taken as the reference image and subtract to every other images.

In order to see the global evolution of the plasma glow, the plasma glow has been recorded

over the whole inter-electrode space. The camera was facing the PKE-Nefedov reactor and

focussed on the centre of the chamber. In Fig.5.23, images extracted from a video (1,789 fps) of

the plasma during a delivery instability are presented. It shows the evolution of the plasma glow

luminosity during one expansion-contraction sequence of the delivery instability (not during the

same experiment as in Figs.5.20 and 5.21). The plasma glow starts to contract toward the centre

before it splits in two luminous vertically aligned glow bubbles. These bubbles tend to come

128


closer to each other before they suddenly disappear and the outer part of the plasma becomes

brighter. The luminosity of the plasma returns then to its original configuration. These effects

are clearly seen in Fig.5.24. In Fig.5.21, it can also be seen that that the void contraction

starts before the central glow faints. The discharge current (Fig.5.24) during the instability also

exhibits interesting features: the first increase corresponds to an increase of luminosity in the

plasma centre. Then the current decreases slowly while the two plasma bubbles are formed and

converge toward each other. The abrupt decrease corresponds to a fast luminosity decrease in

the centre while the plasma edges become brighter. The current decrease indicates that the

total ionisation in the plasma decreases although localized regions of high ionisation exist.

The fact that the enhanced central glow luminosity (suggesting a higher ionisation rate) splits

into two distinctive regions during the delivery instability (Fig.5.23) tends to indicate that there

might be a cloud of very small dust particles in the centre of the void which prevents a high

ionisation rate due to the loss of charge carriers on dust particle surfaces. The complex shapes of

the enhanced ionisation area as well as the presence of this inner dust particle cloud must result

in complex ion drag force and electrostatic force directions and magnitudes. It can however be

deduced that the confining force increases on both dust clouds when the luminous central glow

regions becomes brighter as the void starts to contract before the central high luminosity regions

vanish (Fig.5.21(b)). In the case of the inner growing dust cloud, it is difficult to conclude which

of the ion drag force or the electrostatic force is the dominant force due to the complexities

exhibited by the system. When the situation reverses to dark centre and brighter edges, the

void continues to shrink until it reaches its equilibrium dimension and then a new expansion-

contraction starts again.

(a) (b)

Figure 5.24: Evolution of glow luminosity extracted from the video images presented in Fig.5.23(in false colour from dark red to bright yellow). (a) Evolution of the central column luminosity.(b)Evolution of the central line luminosity (frame rate: 1789 fps). In both cases, the time averagevalue of each pixels has been subtracted in order to enhance luminosity variations. The dashedgreen line superimposed on each picture represents the discharge current (AC component) duringthe delivery instability.

129


As previously discussed, the instability ends with the appearance of a new generation of dust

particles inside the void. Consequently, it can be assumed that some dust particles are growing

inside the void and are modifying the plasma parameters. The evolution rate of the latter must

so be linked to the growth rate of the dust particles. It has been shown in chemically active

plasmas that during the growth of dust particles, the dust particle density ndg decreases at the

beginning when dust particles are growing by agglomeration and then the dust density stays

almost constant when the dust particles are growing by surface molecular deposition [34]. As

the first stage is very short compared to the second one, let’s assume that the density of the

growing dust particles is constant inside the void and that the dust particle radius rdgevolves

as:

rdg(t) = rdmax · (1 − exp(−(t − t0)/τG)) (5.1)

where τG is some growth time constant, rdmax is the maximum radius a dust particle can reach

and t0 the time when the dust particles begin to grow in the discharge. Let’s assume that,

as a first approximation, the void contraction frequency ωd during the delivery instability is a

function of the dust plasma frequency of the growing dust particles ωpdg and the dust plasma

frequency of the old dust particles ωpdo :

ωd = f(ωpdg , ωpdo) (5.2)

If we assume that the properties of the old dust particle cloud remain constant during the

delivery instability, the simplest assumption that can be made is a linear dependence of ωd with

ωpdg and the delivery instability frequency ωd is thus proportional to the dust plasma frequency

of the growing dust particles ωpdg =√

ndgQ2dg

/mdgǫ0 where Qdg is the electric charge on the

growing dust particles and mdg = (4/3)πρr3dg

their mass and ρ their mass density. According to

the OML theory, the dust particle charge is (see Chap.2):

Qdg = Cc · (4πǫ0rdgkBTe/e) ln((ni/ne) · (meTe/miTi)1/2) (5.3)

So it can be considered that in first approximation:

Qdg(t) ∝ rdg(t) (5.4)

Thus it can be deduced that the delivery instability frequency evolves as:

ωd(t) ∝ ωpdg(t) ∝√

r−1dg

(t) (5.5)

The delivery instability frequency has been fitted by such a function and the results is presented

in Fig.5.22. As can be seen, there is good agreement and a growth time constant τg = 2.35s has

been found. This value is in good agreement with the dust particle appearance time measured

in other sputtering experiments [59].

A strong electrostatic repulsion force between the two clouds of dust particles as the major

130


process seems to be, as a first approximation, a good assumption. Consequently, if we assume

that in the ”void” ne ∼ ni (assuming that the density and/or size of the growing dust particles

is not large enough to significantly affect the electron density), by applying Gauss theorem it

can be found that the force exerted on the old dust particle cloud by the growing one Fog evolves

as:

Fog(t) ∝ rdg(t) (5.6)

Thus the amplitude of the void oscillation during the delivery instability has the same evolution.

In Fig.5.22, the delivery instability amplitude has been fitted by a function proportional to the

function rdg(t) obtained by fitting the delivery instability frequency. As can be seen, it matches

well for small amplitude oscillations but some discrepancies appear for larger oscillations. This

was expected as the model neglected other forces such as ion drag force and furthermore, for

big amplitudes non-linear effects have to be taken into account.

Nevertheless, the delivery instability frequency evolution as well as its amplitude evolution con-

firm that the delivery instability is associated with the growth of dust particles inside the void.

These new dust particles cannot be seen by laser light scattering from the beginning of the

instability because they are too small.

5.2.2 The rotating void

(a)

Frame number

Pix

el

nu

mb

er

200 400 600 800 1000 1200 1400

100

150

200

(b)

Figure 5.25: (a) Rotating void: the void is not in the centre of the discharge and can be observedon the right or on the left of the laser sheet alternatively. (b) Line profile during a rotating voidinstability (in false colour from dark red to bright yellow).

In some cases (not well defined yet), the delivery instability evolves into another instability.

The void starts to rotate around the axis of symmetry of the chamber. A typical image of

this situation is presented in Fig.5.25(a). As can be seen, the dust free region is not located

in the centre of the discharge anymore but it is slightly decentred. This behaviour can also be

131


observed on the time evolution of the the plasma glow luminosity. Indeed, in Fig.5.25(b), the

central line profile evolution (i.e. line going through the centre of the discharge) is presented.

The camera was focussed on the centre of the discharge and the frame rate was 1,789 fps. As

can be seen, the bright glow goes from one side of the reactor to the other side. This rotating

glow is associated with the dust void also rotating around the axis of symmetry of the chamber

as the plasma luminosity is most of the time increased inside the void (except for very short

time during heartbeat and delivery instability when the plasma edges are brighter then the void

plasma centre).

5.3 Conclusion

In this chapter, instabilities that can occur when a dense cloud of dust particles is present in the

plasma have been described experimentally. All these instabilities are linked to the presence of

a dust-free void in the centre of the discharge. They can be observed via the discharge current

evolution, the plasma glow evolution and the dust cloud motion. Two types of void instabilities

have been presented:

1. Self-excited instability of the void: the heartbeat instability which can been divided in two

distinct cases: the high-repetition rate heartbeat and the low repetition rate heartbeat.

The last one is characterised by isolated contraction-expansion sequences and the dust-

void has time to reach its equilibrium size before a new contraction occurs while the first

one is characterised by a constant motion of the dust cloud and a constant evolution of

the plasma glow. Intermediate heartbeat cases characterised by the presence of failed or

aborted contractions have also been presented. The asymmetry in the void contraction

that can sometimes occurs was also presented

2. Instability of the void due to the growth of a new generation of dust particles: when

dust particles are growing inside the old dust particle void, it can trigger large amplitude

and low frequency oscillation of the dust void. The amplitude and the frequency of the

void oscillation have been linked to the new generation of dust particles. This instability

can sometimes evolve to a rotation of the dust void around the axis of symmetry of the

discharge.

Thus, the presence of dust particles inside a plasma can lead to unusual instabilities. Theoretical

investigations are needed to fully understand the physical processes involved in the appearance

of these instabilities and the parameters driving their evolution with time.

132



Dans ce chapitre, nous nous sommes interesse aux instabilites d’un nuage dense de poudrespiege dans le plasma. Une region sans poudre, appelee « void »d’environ 1 cm de diametreet aux contours clairement delimites, est souvent presente au centre du plasma. Les theoriesactuelles decrivent communement ce « void »comme decoulant d’un equilibre entre une forceelectrostatique confinante et une force de friction des ions ayant tendance a expulser les poudresdu plasma. Sous certaines conditions, ce « void »devient instable. Ces instabilites peuvent alorsetre reperees sur les parametres electriques de la decharge, sur la luminosite du plasma et sur ladynamique du nuage du poudres. Deux types d’instabilites ont ete observee :

– Le premier type est une instabilite auto-excitee du « void »appelee l’instabilite « heartbeat ».Elle correspond a des battements, c’est a dire des sequences de contraction-expansion(SCE) des dimensions du « void ». Ce type d’instabilite a egalement ete observe lorsd’experiences en microgravite avec des poudres micrometriques injectees. Le « heartbeat »

presente deux cas extremes : un cas avec un taux de repetition faible qui se composede SCE tres rapides separees par de longues periodes durant lesquelles le systeme a letemps de retourner a l’equilibre (ce cas est observe lorsque l’instabilite est sur le point des’arreter), et un cas avec un fort taux de repetition compose de SCE plus lentes mais qui sesuccedent rapidement sans retour a l’equilibre. Dans les deux cas la contraction s’accom-pagne d’une forte augmentation de la luminosite du plasma a l’interieur du « void »(aussiassociee a une forte augmentation du courant) suggerant un accroissement de l’ionisationinduisant une rupture de l’equilibre entre la force electrostatique et la force de friction desions. Quelques millisecondes apres le debut de la contraction du « void », la luminosite aucentre du plasma diminue fortement alors que la peripherie devient brillante. Cela suggereque les regions de forte ionisation se sont deplacees a cause de l’entree des poudres dansle void. Lorsque l’instabilite ralentie avant de s’arreter, elle presente des SCE avorteesentre les SCE completes. Plus le nombre de SCE avortees est grand entre chaque SCEcomplete, plus ces dernieres sont rapides. Il a ete montre que ces SCE avortees suivies deSCE completes sont des « mixed-mode oscillations »[153]. Ces derniers sont connus pourapparaıtre dans les systemes ou differentes echelles de temps coexistent [154].

– Le second type d’instabilite est amorcee par la croissance d’une nouvelle generation depoudres dans le « void ». Il est en effet possibe d’avoir une formation continue de poussieresdans le plasma lorsque les parametres de la decharge sont favorables, et de nombreusesgenerations de poudres peuvent ainsi se succeder. Lorsqu’une nouvelle generation de poudrescroıt dans le « void »de l’ancienne generation, elle peut, sous certaines conditions (non-determinees pour l’instant), declencher une instabilite du « void »qui ne s’arrete quelorsque les poudres de la nouvelle generation ont atteint un taille et/ou une densite cri-tiques. Cette instabilite est marquee par des sequences d’expansion-contraction des di-mensions du « void »a basse frequence et grande amplitude. La difference la plus no-toire avec le « heartbeat »est que dans le cas cette instabilite, les dimensions minimalesdu « void »correspondent a la situation stable. De plus, l’augmentation de la lumino-site au centre du plasma n’est pas homogene : deux regions distinctes de forte lumino-site (suggerant une ionisation plus importante) apparaissent dans les parties superieureset inferieures du « void »suggerant ainsi la presence d’un nuage interne de poudres enformation. L’evolution temporelle de la frequence et de l’amplitude des oscillations est di-rectement reliee a la croissance de la nouvelle generation de poudres. Cette instabilite setransforme parfois en une rotation du void autour de l’axe de symetrie de la decharge.

133

Chapter 6

Complex plasma afterglow and

residual charge

After the studies of the effect of dust particle growth on discharge parameters and the related

instabilities, and the studies of the instabilities of a well established dust particle cloud filling

the interelectrode gap, it is time to investigated the last stage of the life of a complex plasma:

the discharge afterglow. When the power input to the discharge is turned off, the RF field

rapidly disappears and the ionisation processes become negligible compared to the plasma loss

processes. In low pressure monatomic plasmas, plasma losses are essentially diffusion losses and

the charge carriers recombine on the reactor wall. However it is known that, in a complex plasma

afterglow, the dust particles have a strong effect on the plasma loss processes. It has indeed

been shown experimentally that the presence of dust drastically shortens the plasma decay time

[21]. Moreover, as the particles are charged in a running discharge, the dust particle charge has

to relax during the discharge afterglow. The decharging process of the dust particles must be

linked to the diffusion of the charge species. It has also been observed that dust particles can

keep a small residual electric charge in the late afterglow of a dusty plasma [77].

Consequently, in this chapter the influence of dust particles on plasma decay, the dust particle

decharging process and dust particle residual charge have been investigated both theoretically

and experimentally.

6.1 Afterglow in dust free plasma

When a discharge is switched off, the electrons and the ions diffuse to the walls of the reactor.

In a monatomic gas such as argon, the plasma losses are mainly due to diffusion-recombination

at the walls while in molecular gases such as nitrogen, volume recombination must also be

considered. This last process can be neglected in low pressure monatomic gases because it needs

three body collisions which are very unlikely due to small cross sections.

In the following sections, we consider the diffusion of charged species in the dust-free plasma

afterglow of a monatomic gas. Experimental measurements of electron density decay in both

134

6.1 CHAPTER 6. COMPLEX PLASMA AFTERGLOW

nitrogen and argon plasma are presented and compared.

6.1.1 Diffusion in dust free plasma

In most low temperature laboratory discharges, the electron temperature is not in equilibrium

with the ion and neutral atom temperature. Consequently when the discharge is switched off,

the electron temperature relaxes to room temperature following [77, 137]:

dTe

dt= − Te − 1

τT(6.1)

where Te = Te/Ti, τ−1T =

√π/2

√me/mivT i/len

√Te =

√Te/τ∞

T , len is the mean free path of

electron-neutral collision, and τT is the electron temperature relaxation time. The ∞ exponent

stands for the limiting value at very long time.

In the discharge afterglow, ions and electrons diffuse to the wall of the reactor. The ion and

electron density evolution ni(e)(t) are linked to the fluxes of ions and electrons Γi(e):

∂ne

∂t+ ∆Γe = Se (6.2)

∂ni

∂t+ ∆Γi = Si (6.3)

where Si(e) is the ion (electron) source term. In the plasma afterglow, Si(e) = 0 because the

ionization rate is null and there is no sink term. The fluxes of electrons and ions are strongly

linked to the ion and electron densities and the electric field. When there are no external field,

diffusion flux and space charge electric field E are given by::

Γe = −De∇ne − µeneE (6.4)

Γi = −Di∇ni + µiniE (6.5)

where µe(i) = e/me(i)νe(i)n is the mobility of electrons (ions) where νe(i)n is the electron (ion)-

neutral collision frequency, and De(i)/µe(i) = kbTe(i)/e is the diffusion coefficient of electrons

(ions).

The electric field E is determined using the Gauss equation:

∇E =e

ǫ0(ni − ne) (6.6)

At the very beginning of the afterglow, the ion and electron densities are high enough to couple

the diffusion process of the ions and the electrons. Indeed, as the electrons have a diffusion

coefficient which is larger than the ion one, they tend to diffuse faster to the wall of the reactor.

This induces a break in the neutrality condition and creates a space electric field E. This

electric field will slow down the electrons and accelerate the ions. When the plasma density is

high (typically > 108 cm−3 [155]), the electric forces are important and space charge separation,

135


and thus the electric field, will adjust in order that the ions and the electrons diffuse together:

this is the ambipolar diffusion regime. During this regime, ion and electron fluxes are equal

(Γe = Γi = Γ) and have the same ambipolar diffusion coefficient Da. During this regime, we

have ne = ni = n. As Γe = Γi = Γ, Eq.6.4 and Eq.6.5 can be combined giving:

(µini + µene)Γ = −Deµini∇ne − µeµineniE − Diµene∇ni + µeµineniE (6.7)

Thus,

Γ = −(Deµi + Diµe)∇n

µi + µe(6.8)

So, in laboratory plasma where µe ≫ µi the ambipolar diffusion coefficient is (using the definition

of µα and Dα [20]):

Da ≡ Deµi + Diµe

µi + µe≃ Di(1 +

Te

Ti) (6.9)

The solution of Eqs.6.2 and 6.3 can be solved easily for 1D ambipolar diffusion. The density n

of the plasma follows:∂n

∂t− Da∇2n = 0 (6.10)

The variables can be separated, giving:

n(x, t) = X(x)T (t) (6.11)

which when substituted into Eq.6.10 gives:

XdT

dt= DaT

d2X

dx2(6.12)

If Eq.6.12 is divided by XT , the left hand side becomes time dependant only and the right hand

side becomes space dependant only. Consequently both must be equal to a constant we call

−1/τD. It can then be deduced:

T = T0 exp(−t/τD) (6.13)

and for the spatial part:d2X

dt2= − X

DaτD(6.14)

The solution in 1D has the form [20]:

X = A cosx

Λ+ B sin

x

Λ(6.15)

where Λ = (DaτD)1/2 is the diffusion length. Solving Eq.6.14 for a slab geometry of length l

and taking the symmetric lowest-order solution, X and T can be combined to give:

n(x, t) = n0 exp(−t/τD) cos(πx/l) = n0 exp(−t/τD) cos(x/Λ) (6.16)

136


This equation can be generalized to 3D. The diffusion length Λ depends on the geometry of the

reactor. Integrating Eq.6.16 over space, it can be found that in the afterglow the density will

decrease exponentially in time with a time constant τD.

n(t) = n0 exp(−t/τD) (6.17)

When the plasma density is low enough the electrons and the ions start to diffuse with different

speeds and finally diffuse freely. This process is the transition from ambipolar to free diffusion.

Let’s consider that during the transition, the electron flux and ion flux are equal (Γe = Γi = Γ)

but ne 0= ni and ve 0= vi. The hypothesis of proportionality that allows to treat the transition

from ambipolar to free diffusion is [155]:

ni(r)

ne(r)= Ct (6.18)

where Ct is a constant. Eq.6.18 can be rewritten as:

ni∇ne = ne∇ni (6.19)

From Eq.6.7, it is possible to derive Γ:

Γ = −[

(Deµi + Diµe)ni

µini + µene

]∇ne = −Deff∇ne (6.20)

The electric conductivity is:

σ = (µini + µene)e (6.21)

and the charge density is:

ρ = (ni − ne)e (6.22)

Using Eqs. 6.9, 6.20, 6.21 and 6.22, the effective electron diffusion coefficient can be written as:

Deff = Da

(1 +

µeρ

σ

)(6.23)

As can be seen the effective electron diffusion coefficient is directly linked to the conductivity

and the space charge. As Γe = −De∇ne − µeneE = −Deff∇ne, the electric field is:

E = −(De − Deff )

µe

∇ne

ne(6.24)

with the Poisson equation:

∇ · E =ρ

ǫ0= −(De − Deff )

µe∇

(∇ne

ne

)(6.25)

137


and thus:

ρ = −ǫ0(Deff − De)

µe

[(∇ne

ne

)2− ∇2ne

ne

](6.26)

For free diffusion Deff = De and consequently ρ = 0, i.e. the space charge is negligible.

Let’s consider that the plasma is symmetric and derive Deff as a function of the conductivity

σ0 on the axis of symmetry. On this axis, ∇ne = 0 and ∇2ne/ne = −1/Λ2 [155]. Consequently

with Eqs.6.23 and 6.26:

Deff = Da

(1 +

ǫ0(Deff − De)

Λ2σ0

)

= Da

(1 + Λ2σ0

Deǫ0DaDe

+ Λ2σ0Deǫ0

)(6.27)

If we suppose that the conductivity is mainly electronic (σ0 ≈ µenee), then:

Deff ≈ Da

(1 + Λ2

λ2De

DaDe

+ Λ2

λ2De

)(6.28)

The evolution of the electron diffusion coefficient derived from Eq.6.28 is presented in Fig.6.1. As

10−1

100

101

102

103

104

10−1

100

101

102

(

Λ

λDe

)2

Ds

Da

ions (Gerber and Gerardo)

electrons (Freiberg and Weaver)

electrons (Gerber and Gerardo)

electrons (analytic approximation)

ambipolar diffusion

Figure 6.1: Evolution of the electron and ion diffusion coefficient (Ds; s = i, e) as a function of(Λ/λDe)

2

can bee seen in Fig.6.1, the deviation from ambipolar diffusion using the analytic approximation

starts to be important when (Λ/λDe)2 ∼ 1. A common approximation for the transition from

ambipolar to free diffusion is to consider that ambipolar diffusion takes place until (Λ/λDe)2 = 1

and then electrons and ions diffuse freely.

However, experimental measurements in helium discharge have shown that the deviation from

ambipolar diffusion starts when (Λ/λDe) ∼ 100 [128, 156]. The experimental results extracted

from Refs.[128, 156] are presented in Fig.6.1.

138


6.1.2 Measurement of plasma decay in dust-free plasma

Measurements of the electron density decay in afterglow plasma have been performed in two

kinds of dust-free plasma: a molecular gas plasma (nitrogen N2) and a monatomic gas plasma

(argon Ar). The microwave cavity resonance technique (see Sec.3.2.5) was used to measure the

electron density.

In the case of the argon plasma, the Ar flow was 41.8 sccm. The operating pressure was

P = 1.1 mbar and the RF power was PW = 20 W. In order to reconstruct the resonance curves,

the plasma was pulsed with times ton = 10 s and toff = 30 s. These times were necessary to

obtain the same conditions for each run and construct accurate resonance curves.

In the case of the pure nitrogen plasma, the N2 flow was 43.5 sccm. The operating pressure was

P = 0.9 mbar and the RF power was PW = 40 W. In order to reconstruct the resonance curves,

the plasma was pulsed with times ton = 10 s and toff = 30 s.

As can be seen on Fig. 6.2 , the electron density decay is slightly different for nitrogen afterglow

Time (us)

Frequency

(MHz)

Am

plit

ud

e (

a.

u.)

(a)

Time ( s)µ

Am

plit

ude (

a.u

.)

Frequency

(MHz)

(b)

! "!! #!! $!! %!! &!!!&!

!"

&!!&

&!!

&!&

Time (µs)

ne/ne0

'

'

()

*"

(c)

Figure 6.2: (a) Evolution of the microwave cavity resonant frequency in an Ar afterglow plasma.(b) Evolution of the microwave cavity resonant frequency in an N2 afterglow plasma. (c) Evo-lution of the relative electron densities in Ar and N2 afterglow plasmas

139


plasma and argon afterglow plasma. The plasma decay time of the nitrogen afterglow plasma

is found to be τ0LN2

∼ 200 µs at the beginning of the afterglow and evolves to τLN2∼ 350 µs

at the end of the data span. The plasma decay time of the argon afterglow plasma is found to

be τ0LAr

∼ 150 µs at the beginning of the afterglow and evolves to τLAr∼ 550 µs at the end

of the data span. The differences between the decay time are partially due to the difference in

pressures. Moreover for a molecular gas plasma, volume losses of charges species are the most

important process while in a monatomic gas plasma, losses are mainly by diffusion (i.e. diffusion

and recombination on to the walls of the reactor). In an argon plasma, the diffusion time is

inversely proportional to the electron temperature (Eq.6.9), whereas in a nitrogen plasma the

recombination coefficient is α(N+2 ) ∝ T−0.4

e [157], and so the evolution of the decay time is more

pronounced in the case of the argon plasma than the nitrogen plasma.

6.2 Afterglow in complex plasma

In a complex plasma afterglow, the dust particles interact with the ions and the electrons and

thus surface recombinations of the charge carriers occur at the dust particle surfaces. Conse-

quently, the dust particles act as a sink for charge species and sink terms must be added in

Eqs.6.2 and 6.3:

Se = −νedne (6.29)

Si = −νidni (6.30)

where νe(i)d is the dust-electron (ion) collection frequency. The plasma losses are, in complex

plasmas, a combination of diffusion losses and losses by absorption-recombination on dust par-

ticle surfaces.

In the first part of this section, the plasma losses due to the presence of dust particles are inves-

tigated theoretically. Then electron density decay measurement in the afterglow of chemically

active dusty plasmas is presented and compared to dust-free plasma electron density decay. Fi-

nally, the possible use of the plasma loss time measurement as a complementary diagnostic for

dusty plasma is presented and supported by the comparison with available experimental data.

6.2.1 Diffusion in complex (dusty) plasma

When a discharge is switched off, the ions and electrons of the plasma starts to diffuse and the

plasma density decreases as follows:dn

dt= − n

τL(6.31)

where n = ni(e)/n0 and τL the plasma decay time. In a complex plasma, plasma losses are due

to the diffusion of the ions and the electrons toward the wall of the reactor and by recombination

140


onto the dust particle surfaces. Consequently, the plasma decay time τL is:

τ−1L = τ−1

D + τ−1A (6.32)

where τD is the plasma diffusion time and τA the dust particle absorption time. As the dust

particles are charged due to the fluxes of ions and electrons flowing to their surfaces, the dust

particle absorption time will depend on the charge (and surface potential) of the dust particle.

Using the orbital motion limited approach the charge on a dust particle is given by [86]:

dQd

dt= Ji − Je = −πer2

d[nevTee−ϕ − nivTi(1 +

Te

Tiϕ)] (6.33)

where Qd is the dust particle charge, Ji(e) is the ion (electron) flux, rd is the dust particle radius,

ni(e) is the ion (electron) density, vTi(e)the ion (electron) thermal speed, Ti(e) is the ion (electron)

temperature and ϕ is the dimensionless dust surface potential. The latter can be linked to the

dust charge number Zd by:

ϕ =Zde

2

4πǫ0rdkBTe(6.34)

It has been shown that when Ti ≪ Te and (rd/λD)2 ≪ Ti/Te where λD is the Debye length,

the trapped ions play a significant role in the charging process of dust particles [87, 88] and

Eq.6.33 has to be modified appropriately. However, for the conditions we are interested in,

(rd/λD)2 ∼ Ti/Te.

In a complex plasma afterglow, the recombination on dust particle surfaces is directly linked to

the flux of ions onto the dust particles as the dust particles are negatively charged in a laboratory

plasma. Consequently, the dust particle absorption time τA can be written as :

τ−1A = πr2

dndvT i(1 +Te

Tiϕ) (6.35)

In a dust-free plasma (ne = ni), the equilibrium surface potential of an isolated dust particle

can be found by solving dQd/dt = 0 and thus by solving:

(Ti

Te

me

mi

)1/2(1 +

Te

Tiϕ0) exp(ϕ0) = 1 (6.36)

Nevertheless, in a complex plasma, the ion density ni and the electron density ne are different

due to the charging of the dust particles. However, as the plasma is quasi-neutral, the electron

density can be deduced using the neutrality equation:

ne = ni − Zd · nd (6.37)

The ratio of the electron density to the ion density can be thus written as:

ne

ni= 1 − Zdnd

ni= 1 − PH (6.38)

141


where PH = Zdnd/ni is the Havnes parameter. As can be seen from Eq.6.37 and Eq.6.38, for

very low dust density, the electron density is very close to the ion density (ne ≃ ni and PH ∼ 0).

Consequently, the surface potential of the dust particle will be the same as for isolated dust

particles and the dust particle absorption time τA0 is inversely proportional to the dust density:

τ−1A0

= πr2dndvT i(1 +

Te

Tiϕ0) (6.39)

However, when the dust density increases, the electron and ion densities start to deviate from

each other and the Havnes parameter increases. Consequently, as already mentioned in Sec.2.1.3,

solving dQd/dt = 0, the dust surface potential follows:

(Ti

Te

me

mi

)1/2(1 +

Te

Tiϕ) exp(ϕ) = 1 − PH (6.40)

This equation has to be solved numerically and the value of the surface potential explicitly

depends on the dust particle density. This will directly affect the dust particle absorption

time. However,if the dust density is not to high, the deviation of the dimensionless dust surface

potential from the isolated dust particle surface potential will be small so one can write ϕ =

ϕ0 + ∆ϕ. Using a first order approximation, Eq.6.40 can be rewritten as:

(Ti

Te

me

mi

)1/2(1 +

Te

Ti(ϕ0 + ∆ϕ)

)exp(ϕ0)(1 + ∆ϕ)

= 1 − Zdnd

ni(6.41)

From Eq.6.34 and Eq.6.36, it can then be deduced that:

∆ϕ = −(1 +

(Te

Ti

me

mi

)1/2exp(ϕ0) +

4πǫ0rdkBTend

e2ni

)−1

× 4πǫ0rdkBTe

e2

ϕ0nd

ni(6.42)

In a typical laboratory plasma, (Teme/Timi)1/2 exp(ϕ0) ∼ 10−2 and (4πǫ0rdkBTend)/(e2ni) ∼

10−1, thus:

∆ϕ ≃ −4πǫ0rdkBTe

e2

ϕ0nd

ni∼ −Zd0

nd

ni(6.43)

where Zd0 = |Qd0/e| is the charge number of an isolated dust particle. The dust particle

absorption time can be rewritten as:

τ−1A ≃ πr2

dndvT i

(1 +

Te

Tiϕ0

)

−πr2dn

2dvT i ·

Te

Ti

4πǫ0rdkBTe

e2

ϕ0

ni(6.44)

The dust particle absorption time is no longer inversely proportional to the dust density. There

is a term in n−2d which tends to attenuate the effect of the dust particles on plasma losses. Indeed

142


when the dust density is high the dust surface potential is lower as is the dust charge number

Zd. Consequently the ion flux is reduced and the dust particle absorption time is increased

compared to the value it would have if the dust particles can be considered as isolated.

Numerical results

0 5 10 150

5

10

15

20

25

30

!"#$%&'

(#)*

!(+

! ,#!&#$%&'(#-.+

/012341#"451#"6!5017#6886)1

/012#"6!5017#6886)1#$&51#39"69#:;;93%0*:103!+

/012#"6!5017#6886)1#$!4*690):<#53<4103!+

=6>=0>'?(6@

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!(

a)

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%!

&!

"!

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Figure 6.3: Particle absorption time as a function of the dust density. a) With or without takinginto account the dust density effect. b) For different Te/Ti

In this section, exact numerical solutions of Eq.6.40 are presented and compared to the

approximate solution of Eq.6.44.

In Fig.6.3, the absorption frequency τ−1A has been computed for ni = 108 cm−3, Te = 0.3 eV

(close to the experimental value of Ref.[21]) and the dust particle radius is rd = 10 µm. As can be

seen in Fig.6.3(a), for low dust densities the dust absorption frequency τ−1A is increasing linearly

as expected. On the contrary, for high dust density the absorption frequency τ−1A deviates from

the linear evolution τ−1A0

. The exact solution and the first order approximation are in agreement

even though some small discrepancies are observed at high dust density. In Fig.6.3(b), the

exact solution is presented for different ratio Te/Ti. As can be seen, the absorption frequency

τ−1A increases when this ratio increases. Indeed, when the electron temperature is higher than

the ion temperature, ion and electron fluxes are greater at equilibrium (Eq.6.6) resulting in an

increased dust particle absorption frequency.

In Fig.6.4, the ratio τ−1A /τ−1

A0as well as the Havnes parameter PH have been computed for

different dust and ion densities. As can be seen in Fig.6.4(a), the deviation from τ−1A0

is marked

at high dust density and low ion density. In Fig.6.4(b) the latter corresponds to a Havnes

parameter close to 1. For small Havnes parameters (i.e. small dust density and high ion density),

it can be seen that τ−1A /τ−1

A0∼ 1. This confirms that when dust particles can be considered as

isolated inside the plasma the absorption frequency is directly proportional to the dust density

while for high dust density the charge carried by a dust particle is reduced as the dust density

increases and consequently the dust absorption frequency does not evolve linearly with the dust

density.

In a plasma afterglow, the plasma density decreases with time. Consequently the effect of the

143


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/,01

!/2*

3,-01

!/2

45

(b)

Figure 6.4: (a) Ratio of the dust particle absorption frequency taking into effect dust densityeffect τ−1

A relative to the dust particle absorption frequency not taking into effect dust densityeffect τ−1

A0for different values of dust and plasma densities. (b) Havnes parameter PH for different

values of dust and plasma densities.

dust particles can become more and more important over time. In order to see this effect,

the first instant of the decay of a complex plasma has been computed using Eq.6.31, Eq.6.32,

Eq.6.35 and Eq.6.38. The results are presented in Fig.6.5. The plasma diffusion loss is taken into

account by choosing a diffusion frequency τ−1D = 500 Hz (which is a typical value for a laboratory

discharge with an argon pressure around 1 Torr and these plasma parameters). As can be seen in

! "! #!!!

$

%

&

'

#!()#!

'

*+,-).!/0

1+).2,!30

! $! %! &! '! #!!!

"

#!

#"

$!

$"

*+,-).!/0

! 4!#).(#!3)560

178!)2,

!3

1783")#!

3)2,

!3

178&")#!

3)2,

!3

1789")#!

3)2,

!3

178#$")#!

3)2,

!3

(a)(b)

Figure 6.5: (a) Evolution of ion density ni during the first microsecond of a plasma afterglowfor different dust densities. (b) Evolution of the dust particle absorption frequency τ−1

A duringthe first microseconds of a plasma afterglow for different dust densities. The diffusion frequencyis τ−1

D = 500 Hz

Fig.6.5(a), the dust particles enhance plasma losses and the greater the dust density, the greater

the effect on plasma losses. In Fig.6.5(b), it can be seen that at the very beginning of the plasma

afterglow, the absorption frequency is roughly proportional to the dust density. This is due to

the high value of the plasma density at this time which allows a small Havnes parameter and

144


thus dust particles can be considered as isolated. However, as the density decreases, the effect

of the dust particles become more and more important and the particle absorption frequency

decreases. Indeed the Havnes parameter increases with time as the plasma density decreases.

Consequently the dust particles are more and more affected by the presence of the other dust

particles. The ion flux as well as the dust particle charge number decrease leading to a decrease

of the dust absorption frequency. This effect is clearly visible in Fig.6.5(b) for high dust density.

6.2.2 Measurement of plasma decay in complex plasmas

In order, to check the influence of the presence of dust particles in an afterglow plasma, the elec-

tron density decay has been measured for different gas mixtures in chemically active plasmas.

Dust particles have been grown in Ar/CH4 plasma, Ar/SiH4 plasma and N2/CH4 plasma. The

microwave resonant cavity technique (see Sec.3.2.5) was used to measure the electron density.

In the case of the Ar/CH4 plasma, an argon flow of 41.8 sccm and a methane flow of 0.8 sccm

were used. The dilution rate was thus 1.9%. The operating pressure was P =1.1 mbar and the

RF power was PW = 20 W. In order to reconstruct the resonance curves, the plasma was pulsed

with ton = 40 s and toff = 80 s. These times are necessary to obtain the same conditions for

each plasma and achieve good resonance curves.

In the case of the N2/CH4 plasma, an argon flow of 43.5 sccm and a methane flow of 1.3 sccm

were used. The dilution rate was thus 3%. The operating pressure is P =0.9 mbar and the RF

power was PW = 40 W. In order to reconstruct the resonance curves, the plasma was pulsed

with ton = 30 s and toff = 60 s.

In the case of the Ar/SiH4 plasma, an argon flow of 20 sccm and a silane flow of 1.2 sccm were

used. The dilution rate was thus 6%. The operating pressure was P =0.12 mbar and the RF

power was PW = 10 W. In order to reconstruct the resonance curves, the plasma was pulsed

with ton = 0.8 s and toff = 20 s.

The plasma decay times have been estimated just after the RF power supply has been turned

! "!! #!! $!! %!! &!!!&!

!"

&!!&

&!!

&!&

!"#$%&µ'(

)$%&*+,-%.#!/(

'

'

()'*+,-.!/)00'123-43'&5&'463)7()89:

#'*+,-.;'123-437

()8<=:#'*+,-.;'123-437

()'*+,-.!/)00'123-43'!5&"'463)7

(a) Argon plasmas

! "!! #!! $!! %!! &!!!&!

!"

&!!&

&!!

&!&

!"#$%&µ'(

)$%&*+,-%.#!/(

'

'

("')*+,-!./00'123,435

("678

#')*+,-9'123,435

(b) Nitrogen plasmas

Figure 6.6: Electron density decay in dusty and dust-free plasmas.

145


off for the various plasmas (Fig.6.6). It has to be noted that before the discharge is switched off

the electron density is always smaller in the dusty plasma than in the dust-free plasma at the

same pressure.

For the Ar/CH4 plasma (in which 200 nm diameter dust particles are grown), the decay time

is τ0LAr/CH4

∼ 83 µs which is much shorter than the decay time measured for a pristine argon

plasma at the same pressure (τ0LAr

∼ 150 µs). It has also to be noted that in this Ar/CH4

plasma the electron density does not decrease immediately and a small increase of the electron

density is even observed at the very beginning of the afterglow. This effect has also been re-

ported in argon-acetylene discharge [158].

For the Ar/SiH4 plasma (in which 50 nm diameter dust particles are grown), the decay time

is τ0LAr/CH4


plasma at the same pressure (τ0LAr

∼ 60 µs)1. However, no increase of the electron density is

observed at the very beginning of the afterglow.

For the N2/CH4 plasma (in which 150 nm diameter dust particles are grown), the decay time

is τ0LAr/CH4


plasma at the same pressure (τ0LN2

∼ 200 µs). Again, in this case, no increase is observed at the

very beginning of the afterglow.

To conclude, measurements of the electron density decay in various dusty plasmas have shown

that the presence of dust particles drastically shorten the plasma decay time as predicted the-

oretically. However, in one particular gas mixture (Ar/CH4), the electron density decay does

not take place immediately but is preceded by a short and small increase of the electron density.

Such an increase has been also reported in a C2H4/Ar afterglow plasma (in which the dust

density can be very high (nd ≃ 4.5 · 106 cm−3)) [158, 159]. Berndt et al. attributed this increase

to an electron release by the dust particle [158, 159]. Such an increase has already been observed

in pulsed helium discharges and attributed to a re-ionisation due to metastable-metastable col-

lisions [160, 161], and as the presence of high dust particle density in an argon dilution plasma

enhances the metastable density [33]. The fact that this electron density increase is not observed

in other mixtures suggests that this release of electrons may be dependant on the dust particles

material.

6.2.3 Complex plasma afterglow as a diagnostics

As we demonstrated in Sec.6.2.1, the dust density influences greatly the dust absorption-

recombination process in an afterglow plasma. Thus, the dust absorption frequency does not

increase linearly with the density, and for high density it is smaller than the absorption frequency

that would occur if each dust particle could be considered as isolated in the plasma. Moreover,

it was shown in Sec.6.2.1 that if the dust density is not too high, a first order approximation

of the absorption frequency matches very well with the exact solution. In Eq.6.44, there are

only a few parameters to know (Te, Ti, rd, ϕ0 and ni). Consequently, it is possible to use the

1A break can be seen in the curve corresponding to the electron density decay in a pristine argon plasma atP = 0.12 mbar (Fig.6.6). This break is not physical and is due to an experimental problem.

146


measurement of the absorption frequency in the afterglow of a complex plasma in parallel with

other diagnostics to deduce some important parameters of the plasma.

For example, in Ref.[21], the dust particle absorption frequency has been measured in the after-

glow of a linear pulsed discharge. The argon pressure was 1.9 Torr and the density of dust varied

from 0 to 104 cm−3. The dust particles had a mean radius rd ≃ 15µm. It is reported that the

electron temperature was roughly equal to the ion temperature (Te ∼ Ti) [21]. Consequently,

the influence of trapped ions on the dust particle charge can be neglected. In Fig.6.7, experi-

mental results from Ref.[21] are fitted using Eq.6.36 and Eq.6.44. The fit gives Ti ≃ 0.042 eV ,

(Te/Ti)ϕ0 = 4.4 and thus assuming Ti ≤ Te ≤ 5Ti (depending when the measurement has been

performed in the plasma afterglow), it gives 107cm−3 ≤ ni ≤ 5 · 107cm−3. The range for the

ion density and the ion temperature are much lower than those that were supposed in Ref.[21]

but it could be due to the fact that their measurements were performed quite late in the plasma

afterglow and temperature relaxation needed to be taken into account.

! " # $ % &!!

'

&!

&'

"!

()*+,&!

-*./

!-0

! 1!&*+,&!-*230

45/677*8()*9/:*;,<=*>;?@AB?

A5(;8>*75B*+76>*A6C*)@?B*);(?5B:0

DE;6>;B5.8A*75B

Figure 6.7: Fit of experimental results of Ref.[21] using the first order approximation (Eq.6.44)

6.2.4 Conclusion

The influence of the dust density on plasma losses has been studied. It has been shown that

the presence of dust particles can drastically shorten the plasma loss time and that the increase

of the dust absorption frequency does not depend linearly on the dust particle density. Finally,

by comparing our results with existing experimental data, the possible use of absorption fre-

quency measurement as a diagnostic for complex plasma is mentioned. Comparison with existing

experimental data [21] supports this idea.

6.3 Residual dust charge

As the charge carries are lost during the afterglow stage, the charge on dust particles must evolve

at the same time. There are many publications reporting on the investigation of dust charging

147


in discharge plasma [80, 86, 98, 162–167] . However there are only few papers devoted to dust

charging, or decharging to be more specific, in the discharge afterglow. In [168] a diffusion of

fine dust particles (8-50nm) in the afterglow of a dusty plasma has been studied. From the

afterglow decay of dust number density the charged fraction of particle was measured. It was

found that some particles carried a very small residual charge and some were neutral. A model

has been proposed to explain charges on particles in the late afterglow of a dusty plasma. In

microgravity experiments, it was shown that dust particles keep about 2% of the charge ac-

quired in the plasma [77]. Experiments were performed in the PKE-Nefedov reactor [64] under

microgravity conditions on board the ISS. The existence of negative residual charges was shown

and a simple theoretical model that describes data obtained was proposed. The residual charges

were attributed to the presence of an excitation electric field that was used for dust charge mea-

surements. Limitations of the Space Station experiment prevented further investigation of this

hypotheses. Theoretical predictions on an effect of RF plasma parameters on residual charges

has not been verified.

So any new experimental evidence of residual dust charges in afterglow plasma will lead to a bet-

ter understanding of the decharging of complex plasma. Indeed the value and nature of residual

charges after plasma extinction is of great importance. The dust particle charge in afterglow

plasma could induce problem in future single-electron devices where a residual charge attached

to deposited nanocrystals would be the origin of dysfunction. It could make easier industrial

plasma processing reactor decontamination using a specially designed electric field. The residual

charges on dust particles in fusion reactors (such as ITER) could also make the cleaning process

much easier.

Furthermore, complex plasma afterglow provides a unique opportunity to perform measurements

of the dust charge distribution (DCD). In a decaying complex plasma, the dust particle charges

as well as the DCD evolve with the other plasma parameters such as the plasma density and

the electron temperature. In the late stages of plasma decay the dust charge is frozen and does

not change further and the DCD remains unchanged for a long time.

In this section, laboratory measurements of dust residual charges are reported. These exper-

iments have been performed using dust particles directly grown in the plasma which can be

confined in the volume of the reactor after the plasma is turned off by using an upward ther-

mophoretic force which balance the gravity force. The residual electric charge of the dust particle

is deduced by inducing oscillations of the dust particle using a well-known sinusoidal electric

field. Thus dust residual charge distributions have been measured for different experimental

conditions

A model taking into account the transition from ambipolar to free diffusion has been developed

to explain the existence of residual charge and the measured dust residual charge distribution.

148


6.3.1 Evidence and measurement of residual dust charge

Evidence of the existence of residual dust charge

High speed video recording of the laser light scattered by the dust particles during the fall of the

dust particles cloud grown in the PKE-Nefedov chamber with PAr = 1.6 mbar and PW = 3.6 W

has been done for different bias voltages applied to the bottom electrode. The evolution of the

central column intensity is presented in Fig.6.8. The intensity of the light is roughly proportional

to the density of dust particles. As can be seen in Fig.6.8, when no bias is applied the dust cloud

>,? >@? >/?

Figure 6.8: Line profile of a high speed video imaging during the fall of the dust particle cloudin discharge afterglow at P = 1.6 mbar (in false colour from blue to red). (a) No bias on thelower electrode. (b) -10V bias on the powered electrode. (c) +10 V bias on the power electrode

is less dispersed that when a positive or negative bias is applied. This is a strong indication that

the dust particles are charged in the discharge afterglow and are interacting with the applied

electric field. Moreover, the observed spread for both positive and negative bias condition is a

strong indication that positively and negatively charged dust particles coexist in the discharge

afterglow.

Measurement of residual dust charge distribution

For the study concerning residual charges, the top electrode was cooled (Fig. 6.9). An upward

thermophoretic force was applied to dust particles in order to counterbalance gravity [78] when

the plasma is off. To study particle charges, a sinusoidal voltage produced by a function genera-

tor with amplitude ±30 V and frequency of 1 Hz was applied to the bottom electrode. Induced

low frequency sinusoidal electric field E(r, t) generated dust oscillations for dust particles that

have a residual electric charge.

An important effect that must be considered in the estimation of the thermophoretic force is

an influence of the finite volume of gas (Fig.6.10). If the pressure is low enough, the gas mean

free path can become comparable to the length scale of experimental apparatus and the gas can

no longer be treated as a continuous medium. Under such conditions, an additional Knudsen

149


ELECTRODE

ELECTRODE

COOLING SYSTEM

GL

AS

S

GL

AS

S

plasma & particleslaser

sheet

field ofview

Figure 6.9: Schematic of the experimental apparatus

number must be considered [105] KnL = lg/L where lg is the mean free path of buffer gas species

and L is the length scale of the reactor. In this experiment, the length between electrodes is

L = 3 cm giving KnL ∼ 5 ·10−3 which means the gas can be considered as a continuous medium.

The temperature gradient between the electrodes was calculated using the FEMLAB c© code

2 rd

FT

∆

Τ

8

particle

Gas

L

T1

T2

Boundary

Boundary

Figure 6.10: Thermophoresis in finite volume of gas

(steady state analysis of heat transfer through convection and conduction with heat flux, con-

vective and temperature boundary conditions using Lagrange-Quadratic element). The temper-

atures on the electrodes were measured by a thermocouple and used as boundary conditions

for the problem. The contour-plot obtained is presented in Fig.6.11. It shows that the vertical

component of the temperature gradient is constant near reactor centre. The value of the temper-

ature gradient is about 2 K/cm for our experiment. There is also a small horizontal component

of the temperature gradient. This is the reason for the particle drift in horizontal direction.

Such drifts allowed us to resolve particle trajectories and made particle charge measurement

more convenient.

The residual charge measurements have been performed as follows. First the chamber was

pumped down to the lowest possible pressure (base pressure ∼ 2 · 10−6 mbar) and the cooling

system was turned on. After this, argon was injected until the operating pressure was reached,

the discharge was started and particles were grown, forming familiar structures such as a void

(see for example [60]). Then, the discharge was switched off and a sinusoidal voltage was applied

150


Figure 6.11: Temperature profile and gradient

UPPER ELECTRODE

CHARGE MEASUREMENT

USED FOR RESIDUAL

OF THE CAMERA

2

1

FIELD OF VIEW

FALLING DUST PARTICLES

DUST PARTICLES OSCILLATING WITH

THE EDGE ELECTRIC FIELD

Figure 6.12: Superimposition of video frames taken with a large field of view CCD camera.Arrows 1 and 2 represent respectively the vertical and the horizontal components of the tem-perature gradient. Edge effects as well as falling dust particles can be seen.

151


to the bottom electrode.

In the afterglow plasma, the dynamics of dust particles is determined by the temperature gra-

dient and excitation electric field. Fig.6.12 presents a superimposition of images taken after

the discharge had been switched off. There are two different types of motion observable. Dust

particles drift upwards, downwards and to the side due to existing temperature gradients and

they oscillate due to the electrostatic force. It is obvious that the thermophoretic force acts

on any dust particle in chamber, while the electrostatic force acts only on particles that have

charge in afterglow. Thus the presence of oscillating particles (see Fig.6.12) clearly indicates

that dust particles do have residual charges after the discharge has been switched off. Dust

particles oscillating in opposite phases as well as non-oscillating dust grains have been observed

indicating that negatively charged, positively charged and non-charged dust particles coexist

after plasma extinction.

It is worth mentioning that in order to observe dust oscillations the discharge must be switched

off abruptly. It was shown that if the power was decreased slowly until the plasma disappears

there are no residual charges (no oscillations of dust particles in the sinusoidal electric field were

observed). Another interesting fact is that residual charge on dust particles has a long relax-

ation time and does not depend on time over which the excitation electric field was applied.

Dust oscillations were observed for more than one minute after plasma extinction and in both

cases when the function generator was switched on during the discharge or few seconds after the

discharge is turned off.

As can be seen in Fig.6.12, there are dust particles falling after the discharge is switched off.

These particles are too big to be sustained by the thermophoretic force. Other particles drift

horizontally at constant height; this means for these particles the gravity force is balanced by

the vertical component of the thermophoretic force. These particles have been used to measure

residual charges. It is clear from Fig.6.12, that use of a large field of view camera gives us good

pictures of the decaying dusty plasma but it is not suitable for residual charge measurement

because edge effects cannot be neglected. Thus a camera with the small field of view was used

for the charge measurement (see Fig.6.12). The superimposition of images from this camera

is shown in Fig.6.13. These images show clear tracks of dust oscillations allowing dust grain

trajectories to be reconstructed

Particle size (mass) and residual charge measurements are strongly related in this experiment.

Charge, size and mass of the dust particles have to be determined. Considering that dust

particles levitating in reactor at a constant height after plasma extinction are the ones for which

the gravitational force is exactly balanced by the thermophoretic force, the dust particle radius

can be found using Eq.2.37 and Eq.2.38:

rd = − 8

5πρg

ktr

vth∇T (6.45)

The dust particles are assumed to be spherical and mainly made of carbon based on the recent

results for dust particles grown in the PKE-Nefedov chamber by sputtering of a polymer material

152


0 0.5 1 1.5 2 2.5−60

−40

−20

0

20

40

60

time (s)

Z−

posi

tion

(p

ixel

s)

grain1grain 2grain 3grain 4grain 5grain 6

Figure 6.13: Left: Superimposition of video frames 10 seconds after plasma extinction. Dustparticle oscillations can clearly be seen. The temperature gradient has a slight horizontal com-ponent. Therefore, trajectories are in the 2D laser plane. Right: Oscillation of 6 dust grains 10seconds after plasma extinction. Non oscillating dust grain and oscillations of opposite phaseare also observed.

[61]. It has been also shown in Samsonov and Goree [74] that particles are mainly spherical

for carbonized dust particles (such as those in the present study). Bouchoule and Boufendi

showed in silane discharges that when dust particles grow, the shape tends to be spherical [34]

while for sizes of a few tens of nanometres more complex agglomerated structures are observed.

Consequently the mass of the dust particles md can be deduced from Eq.6.45:

md =4

3πr3

d · ρ (6.46)

where ρ is the mass density of graphite. For our experimental conditions, the radius of the

levitating dust grains is estimated to be rd ≃ 190 nm and their mass is md ≃ 6.5 · 10−17 kg.

From the measurements of oscillation amplitude, the residual charge on a dust particle can be

obtained. As the gravitational force is compensated by the thermophoretic force, the equation

of motion for one dust particle, neglecting its interactions with other dust particles, is reduced

to:

mdz = FE(z, t) + Fnd(z) (6.47)

Taking E(t) = E0(zmean) cos(ωt) (the amplitude of the electric field E0 is the one at the mean

dust levitation height zmean) and using Eq.2.27, Eq.2.29 and Eq.6.47, the dust particle oscillation

amplitude b can be obtained [169]:

b(ω, Qd, E0) =QdE0(zmean)

mdω√

ω2 + 4γ2/m2d

(6.48)

where γ = (4/3)√

2πr2dmnnnvTn(1 + αac(π/8)) is the damping coefficient and ω = 2πf where f

is the frequency imposed by the function generator.

153


Eq.6.48 can easily be inverted to yield the dust residual charge:

Qdres =mdb(ω, Qd, E0(zmean))ω

√ω2 + 4γ2/m2

d

E0(zmean)(6.49)

The sign of the dust particle charge is deduced from the phase of the dust particle oscillation

with respect to the excitation electric field. Oscillation amplitudes up to 1.1 mm have been

measured (depending on the operating pressure) and charges from −12e to +6e are deduced

where e is the elementary charge. By measuring the oscillations of many dust particles, dust os-

cillation amplitude distributions have been constructed and were converted to DCDs (Fig.6.14).

The DCD obtained at P =0.7 mbar is difficult to analyse due to the lack of statistics and it can

hardly be compared to the other experimental DCDs because the particle were not of the same

size. It has been found that, for dust particles of similar sizes, at high pressures dust particles

keep a higher mean residual charge (Fig.6.14 and Tab.6.1). The distributions have been fitted

by a Gaussian function (red line). The mean residual charges are negative for every pressures

and their values correspond to a few electrons (Qdres ∼ −3e) for P =0.4 mbar, Qdres ∼ −3e)

for P =0.7 mbar and (Qdres ∼ −5e) for P =1.2 mbar. Positive residual charges were observed

for the every cases. The standard deviation of the DCD is of the same order of magnitude for

every pressures (σ ∼ 2e), see Fig.2. The coefficient δ = σ(Qdres)/√

|Qdres | is about unity.

The error on the measurement need to be discussed. The assumption that dust particles in

the experiment are almost mono-size is based on the fact that only particles of particular size

can be levitated when the discharge is switched off. Taking into account that the temperature

on the electrodes is known with a precision of ±0.5 K, dust particles with radius 190 ± 15 nm

will be levitated under our conditions. In order to avoid as much as possible uncertainties in

relation to dust particle size, measurements have been performed a few seconds after plasma

extinction. Just after the plasma extinction, various sizes of dust particles can coexist. At this

time, the heaviest dust particles (not sustained by the thermophoretic force) fall down, whereas

lightest ones move upwards. Thus, waiting few seconds allows these particles to be eliminated

and restricts size dispersion. Furthermore, measurements have been performed on horizontal

trajectories, meaning that gravity is closely counterbalanced by thermophoresis.

The uncertainty on the neutral drag force is mainly due to the knowledge of the dust particle

radius and the accommodation coefficient αac. In an argon discharge, the accommodation coef-

ficient is often taken to be αac = 1 (see for example Ref.[170]). However, it has been measured

that for carbon particles at room temperature, αac=0.90 [171]. Consequently, the overall preci-

sion on the damping coefficient is about 17%

Taking into account all sources of errors (radius, mass, electric field, neutral drag force), the

residual charge on each particle is known with a precision of ±50%.

If we apply the potential error of the experimental procedure to a theoretically predicted Gaus-

sian distribution of residual charges with δ = σ/√

Qmean = 0.5 and |Qmean| = 5e we will obtain

a distribution with the same Qmean but broadened to δ = 0.75. This value of δ is below the

measured value of δ = 1 for a comparable DCD (pressure =1.2 mbar). For the measurement at

154


0.4 mbar (with a different Qmean), the same procedure will lead to an broadened DCD with δ

changing from 0.5 to 0.80. In this case we are also below the measured value of 0.91.

By addressing the problem in a reverse way (if we consider that the measured DCDs have been

enlarged by the error), we can artificially remove this potential error on the DCDs. For 0.4 mbar

we find an estimated error-free width of δ = 0.72 and for 1.2 mbar an estimated error-free width

of δ = 0.79.

We can also analyse the error that can arise from the limited statistics. It is known that the

relative uncertainty of the variance for m experimental points is ∆σ =√

2/(m − 1) which leads

to ∆σm = 12% for the measurement at 1.2 mbar (and thus an experimental width of δ = 1±0.07

and an estimated error-free width of δ = 0.79±0.09), and ∆σm = 17% for 0.4 mbar (and thus an

experimental width of δ = 0.91± 0.09 and an estimated error-free width of δ = 0.72± 0.11). In

both cases the estimated values of δ is larger than the δ = 0.5 predicted for a running discharge

[56, 79, 90, 93, 94].

Table 6.1: Measured maximum, minimum and mean dust particle residual charges for twooperating pressures.

P = 1.2 mbar P = 0.4 mbar

∇T −190 K · m−1 −177 K · m−1

rd 194 nm 180 nmmd 6.9 · 10−17 kg 5.4 · 10−17 kgγ 1.36 · 10−13 kg · s−1 0.39 · 10−13 kg · s−1

Qdresmax +2e +6eQdresmin −12e −13eQdresmean −5e −3e

−10 −8 −6 −4 −2 0 20

5

10

15

Charge (e)

Num

ber

ofdust

part

icle

s

−15 −10 −5 0 50

10

20

30

40

Charge (e)

Num

ber

ofdust

part

icle

s

−10 0 100

2

4

6

8

Charge (e)

Num

ber

ofdust

part

icle

s

Qd = −5.36e

σ(Qd) = 2.32eσ(Qd)√

|Qd|= 1.00

Qd = −3.47e

σ(Qd) = 1.70eσ(Qd)√

|Qd|= 0.91

P=1.2 mbarP=0.7 mbarP=0.4 mbar Qd = - 2.57e

σ(Qd) = 2.73eσ(Qd)√

|Qd|= 1.71

Figure 6.14: Measured residual dust charge distributions. From the left to the right: P =1.2 mbar with grown dust particles; P = 0.7 mbar with injected rd = 250 nm dust particles;P = 0.4 mbar with grown dust particles.

155


6.3.2 Modelling of dust particle decharging

The evolution of dust charge in afterglow plasma depends on the evolution of plasma parameters.

In a decaying plasma (afterglow plasma), the kinetic of plasma losses is mainly governed by the

electron temperature relaxation, plasma diffusion and recombination processes [137].

A four stage model

The charging (decharging) process of dust particle in a plasma is governed by the contributions

of all currents entering (or leaving) the dust surface, involving plasma electron and ion currents,

photoemission and thermionic emission currents, etc:

dQd

dt=

∑Ia +

∑Il (6.50)

where Ia and Il are currents absorbed and emitted by the particle (with appropriate sign). As

already mentioned, in most cases for discharge plasmas we can ignore the emission current and

kinetics of the particle charge is given by Eq.6.33. According to Eq.6.33, the charge on a dust

particle depends on the electron-ion masses, temperatures and density ratios me/mi, ne/ni,

Te/Ti. Thus to analyse the decharging of dust particle in afterglow plasma one need to consider

the kinetics of plasma decay.

The plasma diffusion loss and electron temperature relaxation determine the kinetics of the

discharge plasma decay [137]. In presence of dust particles, plasma loss is due to diffusion

to the walls complemented by surface recombination on dust particles. Equations for electron

temperature relaxation and plasma density decay are given by Eqs.6.1 and 6.31 [137, 172]. The

expressions for the time scales are [137, 172]:

τ−1L = τ−1

D + τ−1A (6.51)

τ−1D ≃ linvTi

3Λ2(1 + Te) ≡

1

2(1 + Te)

1

τ∞D

(6.52)

τ−1A ≃ πr2

dndvTi(1 + ϕTe) ≡(

1 + ϕTe

1 + ϕ

)1

τ∞A

(6.53)

τ−1T =

√π

2

√me

mi

vTi

len

√Te ≡

√Te

τ∞T

(6.54)

where τD is the ambipolar diffusion time scale onto the walls, τA is the particle absorption

time scale, li(e)n is the mean free path of ion (electron)-neutral collision, Λ is the characteristic

diffusion length (Λ ∼ 1 cm in this experiment). The ∞ exponent stands for the limiting value

at very long time.

For charging time scales less than the plasma decay or temperature relaxation time scales the

charge on dust particle is in equilibrium, i.e. ion and electron fluxes balance each other, ϕ ≃ ϕeq

156


Table 6.2: Values of the different time scales for two operating pressures.


τ0D 90 µs 260 µs

τ∞D 4.5 ms 13 msτ0A 0.4 ms 0.4 ms

τ∞A 30 ms 30 msτ0L 75 µs 160 µs

τ∞L 3.9 ms 9 msτ0T 90 µs 38 µs

τ∞T 900 µs 380 µsτ0Q 4 µs 4 µs

tc 50 ms 110 msτQ(tc) 1.4 s 1.4 s

and using Eq.6.33, ϕeq is given by:

ne

ni

√Tee

−ϕeq =

√me

mi(1 + Teϕeq) (6.55)

In this case, expressions for charge fluctuation and charge fluctuation time scale τQ are:

dQd

dt≃ −Qd − Qdeq

τQ(6.56)

τ−1Q ≃ vTird

4λ2i0

(1 + ϕeq)n ≡ n

τ0Q

(6.57)

where λi0 =√

ǫ0kBTi/n0e2 is the initial ion Debye length.

It should be noted that the time scale for dust charge fluctuations strongly depends on plasma

density and can vary from microseconds at the initial stages of plasma decay up to seconds for

an almost extinguished plasma. Taking into account Eq.6.31 and Eq.6.57, the time dependence

of τQ can be expressed as [77]:

τ−1Q =

1

τ0Q

exp(−t/τL) (6.58)

To understand the dusty plasma decharging dynamics we have to compare different time scales.

In Tab.6.2, time scales for this experiment are presented. It can be seen that the initial charge

fluctuation time scale is the shortest. The temperature relaxation time scale is shorter or becomes

comparable (for 0.4 mbar) to the plasma density decay time scale, and plasma losses are mainly

determined by diffusion. The latter means that for our experimental conditions dust particles

did not affect the initial stages of plasma decay. Fig.6.15 presents the qualitative dependence

of the main plasma and dust particle parameters during the afterglow. Four stages of the dust

plasma decay can be labelled.

157


Qd

n /i

ne

n0 Te / T

in /

in

eTe / T

i

n0

Qd

t ptct

d

1 1

τT

IVIIII II

Time (s)Plasmaoff

Figure 6.15: Qualitative time evolution of dust charge, plasma density and electron temperatureduring the afterglow. Four stages of the dust plasma decay can be identified: I - temperaturerelaxation stage up to tT II - plasma density decay stage up tp, III- dust charge volume stagetc, IV - frozen stage.

1. As we can see the first stage of the plasma decay (t < τT ) is characterized by the electron

temperature Te falling to room temperature, while the plasma density (especially in case

τ0T < τ0

L) is only slightly decreased. As the charging time scale is almost independent of

Te (Eq.6.57), the charge is still determined by its equilibrium value (Eq.6.55). During the

temperature relaxation stage the particle charge should decrease to the value [77]:

QdrT =1

Te0

(ϕeq(1)

ϕeq(Te0)

)Qd0 ≃ Qd0

62≃ −15e (6.59)

where Qd0 is the initial dust charge in the plasma and QdrT the value of the residual dust

charge at the end of the first decay stage. The dust charge in the plasma Qd0 was esti-

mated as Qd0 = −950e by solving numerically Eq.6.55, with given parameters Ti = 300 K

and Te ≃ 3 eV , for argon plasma with ni∼= ne. It has been shown that when Ti ≪ Te and

(rd/λD)2 ≪ Ti/Te where λD is the Debye length, the trapped ions play a significant role

in the charging process of dust particles [87, 88]. However, as the electron temperature

decreases very fast the effect of the trapped ions rapidly becomes negligible and thus is

not taken into account in our model.

2. At the next stage of decay, electron temperature is stabilised while the plasma density is

still decreasing (see fig 6.15). So τQ continues increasing according to Eq.6.33 and Eq.6.57.

When τQ becomes comparable to τL, the particle charge can not be considered to be in

equilibrium and to determine particle charge we should use Eq.6.33. The time scale when

the particle charge starts sufficiently deviating from the equilibrium value can be estimated

158


as (Eq.6.51 and Eq.6.57):

td ∼ −τ∞L ln

(8

3·(λi0

Λ

)2· lin

rd

)∼ 7τ∞

L (6.60)

However, according to Eq.6.33, as long as plasma is neutral (ne = ni) the charge on dust

particles does not change. The plasma will maintain quasineutrality until the decay rates

for the electrons and ions are the same. This will be the case for ambipolar diffusion.

When the nature of the diffusion changes, electrons and ions start to diffuse independently

resulting in a changing ne/ni ratio and consequently a changing dust charge change.

3. The nature of the plasma diffusion changes when the particle volume charge cannot be

ignored or when the plasma screening length becomes comparable to the chamber size.

In first case the ion diffusion will be influenced by the negatively charged dust particles,

while the electrons will diffuse freely. In second case, large density differences appear over

distances less than the screening length and electrons and ions diffuse independently. Lets

us estimate the characteristic times for both cases.

The influence of the overall particle charge is determined by the value of the Havnes

parameter PH = −ndQd/ene. Based on the model discussed, the qualitative evolution

of the Havnes parameter PH in the dusty plasma afterglow can be plotted as shown

in Fig.6.16. The initial value of PH is small (∼ 0.06 with an estimated dust density

nd ∼ 2 · 105 cm−3 and n0 ≃ ne0 ∼ 5 · 109 cm−3) and there is no influence from dust. At

the first stage of decay (temperature relaxation stage) PH decreases due to a dramatic

decrease of dust charge while the plasma density decreases by a factor of 1.1. At t = τT ,

PH reaches its minimum value of ∼ 10−3. After this, PH starts increasing. During this

stage the dust particle charge changes slowly while plasma number density decays quickly

(see Fig.6.15). The time at which PH becomes ∼ 1 can be estimated as (Eqs.6.31-6.54):

tp ∼ τ∞L ln

((Te0

Tn

)(−en0

QdN

))∼ 8τ∞

L (6.61)

The screening length becomes comparable to the chamber size, i.e. λi(nc) ∼ Λ when the

density drops down to nc = λ2i0/Λ2. This occurs at [77]:

tc ∼ τ∞L ln n−1

c ∼ 12τ∞L (6.62)

At time t = min[tp, tc], electrons start running away faster than ions and the ratio ni/ne

grows. For our experimental conditions tp < tc, thus the neutrality violation due to the

presence of dust particles happens before the Debye length exceeds the chamber size. So the

third stage of dusty plasma decay starts at tp. During this stage the charge on dust particles

changes due to the changing ne/ni ratio. At this stage td < tp, thus Eq.6.33 should be used

for estimating the charge variation. The upper limit of the charge change can be estimated

159


Time (s)

! t

I II III

HminP

H0P

PH

1

T p

Figure 6.16: Qualitative time evolution of the Havnes parameter after the power is switched off.

by ignoring the electron current and considering the time interval between tp and tc,

dQ

dt< Ji

tc

tp(6.63)

< πer2dni(tp)vTi

(1 − e

4πǫ0kBrdTiQd

)tc

tp(6.64)

Solving Eq.6.63, the charge should evolve following:

Qd =(QdT − 1

α

)exp

(− Kα∆t

)+

1

α(6.65)

where α = e/4πǫ0kBrdTi ∼ 0.28/e, K = πer2dni(tp)vTi ∼ 190e, and thus (Kα)−1 ∼

20 ms ∼ ∆t = (tc − tp).

Therefore, the charge during the third stage decreases to −3e.

4. At the forth stage of plasma decay, t > tc, the plasma density decreases such that any

further changes of dust charge become negligible and charge remains constant for a while.

Thus the final residual charge for our condition is expected to be about Qdres ∼ −3e which

is well correlated with the charges measured in the experiment.

This model predicts the existence of a negative residual charge of a few electrons but it is

unable to predict a positive residual charge. The limits of the model are the step transition

from ambipolar to free diffusion and the failure to properly take account of the influence of dust

particles. Indeed the effect of the dust particles is ignored until the dust charge density is the

same as the electron density, which leads to an abrupt change from ambipolar to free diffusion

of ions and electrons. The assumption of the step transition from ambipolar to free diffusion is

quite unphysical and contradicts existing experimental data [128, 156, 173, 174]. As previously

mentioned, electrons and ions are diffusing at different rates very early in the decay process.

This smooth transition must lead to a change in dust charges as the ratio ni/ne will be modified

early in the decaying process. Under the conditions of our experiments, the ratio (Λ/λDe) ∼ 50

when the discharge is switched off. Consequently, this must result in a different dust particle

160


charge evolution in the afterglow compared to previous assumptions [77, 92]. For this reason, in

the next sections, a modified model taking into account the gradual transition from ambipolar

to free diffusion is used to simulation of complex plasma afterglow.

A Fokker-Planck model

In plasma afterglow, the electron temperature relaxes due to energy exchange in collisions with

neutrals. At the initial stage of plasma decay the electron temperature Te0 is much higher than

the neutral temperature T . In our model, the electron temperature Te decreases during the

afterglow period and tends asymptotically to T (Eq.6.1).

In the presence of dust particles in the plasma, the temperature relaxation and diffusion losses

may be different from those for a dust-free plasma. However, no experimental data nor the-

oretical estimations of the transition from ambipolar to free diffusion in dusty plasma exist.

Consequently, in our model, we will assume that the electron temperature relaxation and diffu-

sion processes of ions and electrons are not significantly affected by the presence of dust particles,

allowing us to treat ion and electron diffusion in a similar way to dust-free plasma. In the model,

electron and ion density diffusion losses are thus treated separately and follow:

dni

dt= − ni

τDi(6.66)

dne

dt= − ne

τDe(6.67)

where ni(e) is the ion (electron) density and τDi(e)= Λ2/Di(e) is the ion (electron) diffusion time.

In the simulation, electron and ion densities are studied in a fluid manner solving numerically

Eqs.6.66 and 6.67 using the same constant time step tf << τDi(e)0where τDi(e)0 is the diffusion

time for ions (electrons) at the beginning of the simulation (i.e the very beginning of the decaying

plasma).

The diffusion times were computed taking into account experimental data on the transition from

ambipolar to free diffusion and are a function of the ratio (Λ/λDe)2 and are calculated using

experimental results from Gerber and Gerardo [156] or Freiberg and Weaver [128] (Fig.6.1).

These data give the ratio of ion and electron diffusion coefficients to the ambipolar diffusion

coefficient Da as a function of the ratio (Λ/λDe)2 in helium plasma where λDe is the electron

Debye length. The ambipolar diffusion coefficient is:

Da ∝ ((kBTi)3/2)/(Pσinm

1/2i ) (6.68)

where P is the neutral gas pressure and σin is the ion-neutral collision cross section. As

σinAr+ ∼ 2.5σinHe+[106] and miAr+ ∼ 10miHe+

, DaAr ∼ (1/8) ·DaHe for equal pressures. Conse-

quently, it can be assumed that the diffusion of an argon plasma with argon pressure PAr has the

same behaviour as an helium (He) plasma with helium pressure PHe ≃ 8 · PAr. The available

161


experimental data for Helium are in the range of pressure 0.4− 4 Torr [128] and 9− 22.8 Torr

[156] which correspond to 0.05− 0.5 Torr and 1.13− 2.85 Torr respectively in equivalent argon

pressure. Our simulations were performed in the range 0.3 − 1 Torr (0.4 − 1.3 mbar) Conse-

quently, the data from Ref.[128] are used to compute the diffusion time of electrons and data

from Ref.[156] are used for ions as they are only data available.

As the diffusion is treated as in a dust-free plasma, it restricts the validity of our simulations

to the case of low dust particle densities. In this case, the influence of dust particles will be re-

stricted to plasma absorption losses on the particle surface. Indeed, when immersed in a plasma,

a dust particle acquires a net electric charge due to ions and electrons ”falling” on its surface

[5, 83, 86]. In dusty plasmas, ions and electrons captured by dust particles can be considered as

”lost” by the plasma and thus the charging process of dust particles is also a loss process.

A dusty afterglow plasma is not a steady case but the OML approach can be used to obtain

the charge on dust particles. A Fokker-Planck model of dust charging due to the discreetness of

the charge currents (ion and electron) can be used to obtain dust particle charge distributions.

Indeed, the plasma particle absorption time interval as well as the sequence in which electrons

and ions arrive at the dust particle surface vary randomly but observe probabilities that depend

on the dust particle potential φd. The probability per unit of time for absorbing an electron

pe(φd) or an ion pi(φd) are calculated from the OML currents [93]:

pe = −Ie/e (6.69)

pi = Ii/qi (6.70)

The ion and electron currents can be calculated using Eq.2.11 and Eq.2.12 (the equation used

for the selected species depends on the charge of the dust particle). The effect of trapped ions

is neglected as the electron temperature decrease very fast during the afterglow.

The contribution of dust particles to plasma losses is taken into account in the following way:

The charges of Nd dust particles are computed. Knowing the dust density nd in the plasma, an

equivalent volume Veq to these Nd dust particles can be calculated.

For a dust particle, a time step tpj for which the probability of collecting an ion or an electron

of the plasma is computed [93]:

tpj = − ln(1 − R1)

ptot(6.71)

where 0 < R1 < 1 is a random number and ptot = pe + pi is the total probability per unit of

time to absorb an ion or an electron.

While∑

j tpj ≤ tf , an electron or an ion is absorbed during the time step tpj and the nature

of the absorbed particle is determined comparing a second random number 0 < R2 < 1 to the

ratio pe/ptot. If R2 < pe/ptot, then the collected particle is an electron otherwise it is an ion.

The probability pe and pi are recalculated and a new time step tpj+1 is computed.

When∑

j tpj > tf , one more iteration is applied to the dust particle. Nevertheless, the

probability of collecting an ion or an electron in the time interval ∆t = tf −∑j−1

k=1 tpkis

162


p(∆t) = 1− exp(−∆t · ptot) (if tp1 > tf then p(tp1) is computed). A random number 0 < R3 < 1

is then generated and if R3 < p(∆t) a plasma particle (ion or electron) is absorbed. The choice

between ion and electron is decided as previously described.

When all the Nd dust particles have been treated, the number of absorbed ions and electrons

Niabs and Neabs respectively by the Nd dust particles during the time step tf is known and can

be transformed using the equivalent volume Veq into absorbed ion and electron densities niabs

and neabs respectively which are subtracted from the ion density ni and electron density ne

before the next iteration of time step tf .

For each iteration, we calculate the mean time tp necessary for one particle to collect a plasma

particle (i.e. an ion or an electron) as well as the ratio rloss = ne(i)diff/ne(i)abs

where ne(i)diff

is the density of electron (ion) lost by diffusion. The program is stopped when tp >> τDi (

tp > 10 · τDi in this simulation) when the charge on dust particles can be considered as frozen

due to a plasma loss time becoming less than the particle charging time.

The numerical dust particle charge distributions are reconstructed by simulating the charge

of Nd = 500 dust particles corresponding to a density nd = 5·104 cm−3. The initial ion density is

ni0 = 5·109 cm−3 and the initial dust particle charge distribution is computed using a Cui-Goree

algorithm [93] and the quasi-neutrality condition:

Zdnd + ne = ni (6.72)

The initial electron density ne0 is deduced from this calculation. Many iterations of the algorithm

are necessary to obtain the initial dust charge distribution and the initial electron density: the

first iteration assumes ne0 = ni0 = 5 · 109 cm−3, and for the next iteration ne0 is calculated

using Eq.6.72 and the dust particle charges of the first iteration. A new dust particle charge

distribution is then computed. This process is performed again and again until ne0 and the dust

charge distribution are stabilised. The obtained dust particle charge distribution is presented in

Fig.6.17 The mean charge is Qmean ≃ −952e and the standard deviation σ(Qd) ≃ 17e. Eq.6.55

!!""" !#$" !#%" !#&" !#'" !#"""

!"

'"

("

&"

)"

%"

*"

Charge (e)

Num

ber

ofdust

part

icle

s

Qd = −952e

σ(Qd) = 17eσ(Qd)√

|Qd|= 0.55

Figure 6.17: Dust charge distribution for 190 nm radius dust particles with ni0 = 5 · 109 cm−3

and nd = 5 · 104 cm−3

163


predicts dust particle charges Qd = −950e which is in very good agreement with our simulation

results.

The decay of a dusty argon plasma is then simulated using the algorithm previously described

for two gas pressures (P = 0.4 mbar and P = 1.2 mbar (Fig.6.18)).

100

101

102

103

104

105

0

50

100

Time (µs)

Te/T

i

a)

100

101

102

103

104

105

105

107

109

Time (µs)

Den

sity

(cm−

3)

ions

electrons

Zdnd

b)

100

101

102

103

104

105

0

10

20

30

Time (µs)

Dif

fusi

on

tim

e (

ms)

ions

electrons

c)

Figure 6.18: Decay of an argon plasma at P = 1.2 mbar with a fast ambipolar-to-free diffusiontransition. a) Electron temperature relaxation; b) density evolution; c) evolution of diffusiontime.

The ion-neutral mean free path is calculated using the cross section from Varney [106] and

the electron-neutral mean free-path is calculated using the cross section from Kivel [175]. The

initial electron temperature is taken as Te0 = 3 eV and the ion temperature is assumed to be

equal to the neutral temperature Ti = T = 0.03 eV. The diffusion length is taken Λ = 1 cm

which is approximately the diffusion length of the PKE-Nefedov reactor in which experiments

on residual dust charge have been performed [77, 92]. The transition from ambipolar to free

diffusion is based on either experimental results from Gerber and Gerardo [156] or experimental

results from Freiberg and Weaver [128] (the former suggest a slower transition from ambipolar

to free diffusion than the latter, see Fig.6.1).

As can be seen in Fig.6.18, the first decrease of the dust particle charge corresponds to the

electron temperature relaxation. Then, while electrons and ions diffuse ambipolarly, the charge

remains constant. Finally, when the transition occurs (after tens of ms), electron and ions

densities deviate from each other and the dust charge decreases until it freezes.

For a pressure P = 0.4 mbar (P = 0.3 Torr), the simulated final dust charge distributions

(i.e. tp > 10 · τDi) are presented in Fig.6.19. The residual charge is Qdres ≃ −16e when no

transition in the diffusion process is taken into account (Fig.6.19a). When using a model based

on an abrupt transition from ambipolar to free diffusion at a Havnes parameter of PH = 0.5, the

residual charge is smaller (in absolute value) Qdres ≃ −13e but still far from experimental value

(Fig.6.19b). It should be noted that, typically, the Havnes parameter reaches 0.5, after 10 ms;

it corresponds to a Λ/λDe ratio close to unity. Using Gerber and Gerardo data (slow transition,

lower curve in Fig.6.1) , the mean residual charge Qdres ≃ −13e(see Fig.6.19c) whereas it is

164


−20 −15 −10 −50

20

40

60

80

100

Charge (e)

Num

ber

ofdust

part

icle

s

−20 −18 −16 −14 −12 −10 −8 −60

20

40

60

80

100

Charge (e)

Nu

mb

er

of

du

st p

art

icle

s

−8 −6 −4 −2 0 2 40

20

40

60

80

100

120

Charge (e)

Num

ber

of

dust

part

icle

s

−25 −20 −15 −100

20

40

60

80

100

120

Charge (e)N

um

ber

ofdust

part

icle

s

a) b)

c) d)

Figure 6.19: Numerical results for 190 nm radius dust particles with argon pressure P =0.4 mbar (P = 0.3 Torr) and nd = 5 · 104 cm−3. a) Ambipolar diffusion until the end of thedecay process. b) Abrupt transition from ambipolar to free diffusion when PH = 0.5. c) Usingdata from Gerber and Gerardo for transition from ambipolar to free diffusion [156]. d) Usingdata from Freiberg and Weaver for transition from ambipolar to free diffusion [128]

Qdres ≃ −1e (see Fig.6.19d) using Freiberg and Weaver data (fast transition, higher curve in

Fig.6.1). In the slow transition case, there are no positive particles observed in the simulated

dust particle charge distribution whereas there are in the case of the fast transition. The residual

charge distribution for the fast transition is similar to the experimental results (see Fig.6.14).

For a pressure P = 1.2 mbar (P = 0.9 Torr), the simulated final dust particle charge distributions

are presented in Fig.6.20. The residual charge is Qdres ≃ −16e when no transition in the diffusion

process is taken into account (Fig.6.20a). When using a model based on an abrupt transition

from ambipolar to free diffusion (at a Havnes parameter of PH = 0.5), the residual charge is

smaller (in absolute value) Qdres ≃ −7e but still far from experimental value (Fig.6.20b). A

dependence on the ambipolar to free diffusion transition is again seen with Qdres ≃ −14e for

the slow transition and Qdres ≃ −2e for the fast transition. The latest charge distribution is the

closest to experimentally measured distribution (Fig.6.14). The results for both pressures are

summarised in Tab.6.3

In Fig.6.21, the simulated DCDs at different time in the plasma afterglow are presented. The

resulting mean dust charge evolution is in agreement with qualitative predictions of the four

stage model (Fig.6.15).

During the first stage of the plasma decay, the charging time of dust particles tQ0 is very short

compared to the plasma loss time τL (typically tQ0 ∼ 1 − 10 µs << τL ∼ 1 − 10 ms) so the

charge on dust particles can be considered to be in equilibrium with the surrounding plasma. The

electron temperature relaxes to the room temperature leading to a strong decrease of the dust

charge. In Fig.6.21, this corresponds to the first milliseconds. Then during the plasma decay

165


−22 −20 −18 −16 −14 −12 −100

20

40

60

80

100

120

Charge (e)

Num

ber

ofdust

part

icle

s

−15 −10 −5 00

10

20

30

40

50

60

70

80

Charge (e)

Num

ber

ofdust

part

icle

s

−20 −15 −10 −50

50

100

150

Charge (e)

Num

ber

of

dust

part

icle

s

−8 −6 −4 −2 0 2 40

20

40

60

80

100

Charge (e)

Num

ber

ofdust

part

icle

s

a) b)

c) d)

Figure 6.20: Numerical results for 190 nm radius dust particles with argon pressure P = 1.2 mbar(P = 0.9 Torr) and nd = 5 · 104 cm−3.a) Ambipolar diffusion until the end of the decay process.b) Abrupt transition from ambipolar to free diffusion when PH = 0.5. c) Using data from Gerberand Gerardo for transition from ambipolar to free diffusion [156]. d) Using data from Freibergand Weaver for transition from ambipolar to free diffusion [128]

Table 6.3: Dust particle residual charges for two operating pressures.


Experimental results −5e −3eAmbipolar diffusion −16e −16eHavnes transition −7e −13e

Slow transition [156] −14e −13eFast transition [128] −2e −1e

166


!!"# !!$# !!%##

&#

'#

(#

)*+,-.

/*

t=0 µs Qd = −952e

σQd= 17e

δ = 0.55

!0## !!# !"##

$#

0##

0$#

&##

)*+,-.

/*

t=250 µs Qd = −89e

σQd= 4.62e

δ = 0.49

!'% !%( !&!#

$#

0##

0$#

&##

&$#

)*+,-.

/*

t=500 µs Qd = −37e

σQd= 2.92e

δ = 0.48

!&( !&# !0'#

0##

&##

%##

)*+,-.

/*

t=1 ms Qd = −20e

σQd= 2.08e

δ = 0.47

!&0 !0( !00#

0##

&##

%##

)*+,-.

/*

t=15 ms Qd = −16eσQd

= 1.88e

δ = 0.47

!" !& '#

0##

&##

%##

'##

)*+,-.

/*

t=110 ms Qd = −1.74eσQd

= 1.86e

δ = 1.41

Figure 6.21: Simulated DCDs at different times in the afterglow plasma (P = 1.2 mbar and∑Nd = 1500).

stage dust charge stabilizes. During the third stage charge decreases until it becomes frozen.

The simulation is stopped during the fourth stage when the charging time becomes much bigger

than the diffusion time. Drastic DCD broadening is observed in the third stage when the charge

of the particles continues to decrease (from t = 15 ms in Fig.6.21) and δ is steeply increasing.

The broadening of the distribution was observed until the end of the simulation. As can be seen

in Fig.6.21 (t = 110 ms), the frozen DCD is very wide with a tail in the positive charge region.

The simulated DCD is in good agreement with the experimental one (Fig.6.14).

We now discuss the results and limits of our model. In general, we found that the transition

from ambipolar to free diffusion plays a major role in the decharging process of the dust particle.

It is instructive to see how the transition affects the residual dust charge, in particular, the width

of dust charge distribution. The fact that electrons and ions began to diffuse independently at

an early stage of afterglow changes the dust charge distribution function drastically.

As we can see from data in Fig.6.19(a,b,c) for the pressure of 0.4 mbar the distributions for

ambipolar diffusion, abrupt and slow transitions do not differ significantly. So any difference

in diffusion in the late afterglow (low ratio of Λ/λDe) has little effect on residual charge. In

contrast, a slight difference at an early stage has a large influence on the final dust charge dis-

tribution and leads to the appearance of positively charge particles (Fig.6.19d). For the higher

pressure (1.2 mbar) this effect is not so pronounced, and the diffusion in the late afterglow plays

a noteworthy role. In this case an abrupt transition, due to the influence of the dust charge

167


particle volume effect, tends to decrease the residual dust particle charge (Fig.6.20b). It does

not, however, give us the experimentally observed value of residual charge or the positive tail of

the charge distribution. So the results obtained let us conclude that the four stage model can

be used only for the rough estimation of residual charge. For a more accurate calculation one

has to take into account the actual diffusion rates for the electrons and ions.

We now discuss the validity of the model. This model is valid only for low dust particle densities.

As the losses by diffusion are treated as in a dust-free plasma, the influence of dust particle has

to be very small compared to the total process. Indeed if losses due to dust particles are similar

to or greater than those due to diffusion, this last process must be significantly affected. It has

thus been shown that the presence of a high density of dust particles significantly reduces the

plasma loss time [21]. Moreover, if the total dust charge is not negligible compared to that of

ions and electrons, the diffusion process must also be modified as the dust particles will repel

the electrons and attract the ions.

Consequently, the influence of dust particles can be treated independently from the other loss

processes only if the total charge carried by the dust particles is small compared to the charge

carried by electrons or ions during the decay process. Furthermore, losses on dust particle sur-

faces must not be the main loss process (i.e. rloss ≫ 1) to allow ion and electron diffusions to

be treated in the same way as in dust-free plasma. This condition is definitely satisfied for a

discharge plasma but could change during the afterglow so we have to calculate evolution of the

ratio nidiff/niabs

in discharge afterglow where nidiffis the ion density lost by diffusion and niabs

the ion density lost by absorption onto the dust particle surfaces. Fig.6.22 shows that this ratio

stay more or less above 5 during the whole decay process regardless of the conditions of pressure

or data used to take into account the transition from ambipolar to free diffusion (only results

using Gerber and Gerardo data are presented in fig.6.22). It means that during the plasma

100

101

102

103

104

0

5

10

15

Time (µs)

nid

if

f/n

ia

bs

raw data

smooth data

a)

100

101

102

103

104

0

5

10

15

Time (µs)

nid

if

f/n

ia

bs

raw data

smooth data

b)

Figure 6.22: Evolution of nidiff/niabs

using data from Gerber and Gerardo for transition fromambipolar to free diffusion [156]. a) P = 0.4 mbar. b) P = 1.2 mbar

decay, losses to dust particles are very small compared to the losses to the reactor walls and our

model for plasma decay is valid to the end of the afterglow. Furthermore, looking at the ratio

|Qdnd/eni| presented in Fig.6.23, it can be seen that, except at the very end of the afterglow

168


(i.e. a few ms before the residual charges freeze), the total dust charge volume is much smaller

than the total charge of ions. The charge on the dust particles thus represents a negligible part

of the total charge during the major part of the decay process.

The data presented in Figs.6.23(a) and 6.23(b) confirmed that, in our case, the overall influence

of dust particles on the plasma decay can be neglected and, consequently, the assumption of

dust-free plasma diffusion can be considered as valid for the whole decay process.

It should be noted that the transition from ambipolar to free diffusion is not the only process

that influences the dust residual charge. Indeed, it has been shown that in reactive plasmas such

as C2H4/Ar plasma [158, 159] in which the dust density can be very high (nd ≃ 4.5 · 106 cm−3)

[159]), the afterglow electron density shows an unexpected increase at the very beginning of the

decay process. We obtained a similar result in CH4/Ar plasma (Fig.6.6). Berndt et al. at-

tributed this increase to an electron release by the dust particle [158, 159]. However, as such an

increase has already been observed in pulsed helium discharges and attributed to a re-ionisation

due to metastable-metastable collisions [160, 161], and as the presence of high dust particle den-

sity in an argon dilution plasma enhances the metastable density [33], the reason of the increase

of the electron density at the very beginning of the afterglow is still not clear. In all cases, these

”extra electrons” must affect the charge on dust particles and consequently the residual charge

distributions.

Moreover, it has been shown that the pressure in noble gases such as argon and neon significantly

influences the diffusion coefficient [176]. Indeed, at low pressure the electron diffusion cooling

process, i.e. the fast loss of energetic electrons to the walls of the reactor which are imperfectly

compensated through elastic collisions of the electrons with gas atoms resulting in an electron

temperature below that of the gas, causes a reduction of DaP and thus an enhancement of the

ambipolar diffusion time τD = Λ2/Da. As mentioned previously, the dust particle charges are

related to ion and electron density. Diffusion cooling can also affect dust particle charge distri-

bution evolution. Nevertheless, the present model does not take into account the phenomena of

diffusion cooling (see Eq.6.1) and its importance has to be investigated.

101

102

103

104

10−2

100

Time (µs)

∣ ∣ ∣

Qdn

d

en

i

∣ ∣ ∣

Slow transition [27]

Fast transition [24]a)

101

102

103

104

105

10−4

10−2

100

102

Time (µs)

∣ ∣ ∣

Qdn

d

en

i

∣ ∣ ∣


Fast treansition [24]b)

Figure 6.23: Evolution of |Qdnd/eni|. a) P = 0.4 mbar. b) P = 1.2 mbar

169


6.4 Conclusion

In this chapter, the last stage of the life of a complex plasma discharge has been investigated.

The influence of the dust particles on the complex plasma decay has been discussed.

It has been shown experimentally and theoretically that the dust particles can drastically shorten

the plasma decay time as they act as an absorption-recombination sink for the charge carriers

of the plasma. It has been demonstrated that the dust absorption frequency does not increase

linearly with the dust density due to the increase in dust particle-dust particle interactions

when the dust density rises. Under certain conditions, the decay time measurement can be

used as a complementary diagnostics for complex plasmas. This last statement is supported by

comparison with available data.

Moreover, It has been found that the dust particles do retain a small residual charge. Complex

afterglow plasmas offer therefore the unique opportunity to measure the residual dust charge

distribution. The mean residual charge is negative and the dust charge distribution has a tail

that extends into the positive charge region. It is found that the width of the residual dust charge

distribution is larger than the width of the dust charge distribution predicted theoretically for

running discharges. Simulations have shown that the residual dust charge distributions are

very sensitive to the transition from ambipolar-to-free diffusion. The experimental dust charge

distribution is explained by a transition starting early in the plasma decay stage.

170



Ce chapitre est consacre a l’etude de la phase post-decharge des plasmas poussiereux.Dans un plasma sans poudre, lorsque l’alimentation de la decharge est coupee, le taux d’ioni-sation diminue fortement et devient rapidement negligeable. Les porteurs de charge du plasmadiffusent et se recombinent en gaz neutre sur les parois du reacteur. La diffusion des especeschargees se fait selon l’un des deux regimes suivant : la diffusion ambipolaire durant laquelle lesions et les electrons diffusent a la meme vitesse car la charge despace est suffisament importantepour maintenir un champ electrique ambipolaire qui accelere les ions et ralentit les electrons, etla diffusion libre durant laquelle les ions et les electrons diffusent independament. La transitiond’un regime a l’autre se fait de maniere progressive et est fonction du rapport entre la longueurde diffusion du reacteur et la longueur de Debye electronique.Dans un plasma poussiereux, les poudres, chargees de par les interactions avec les ions et leselectrons du plasma, jouent un role significatif dans la disparition du plasma. Elles agissentcomme des pieges sur lesquels les porteurs de charge se recombinent et ainsi reduisent forte-ment le temps d’extinction du plasma. Cet effet a ete mis en evidence experimentalement encomparant les temps d’extinction dans des decharges avec ou sans poudres. En revanche, il estdemontre que le frequence d’absorption des porteurs de charge par les poudres n’augmente paslineairement avec la densite de poussieres. Ceci est du au fait que les interactions poudre-poudre,augmentant avec la densite de poudres, reduisent la charge electrique de ces dernieres et doncles flux de porteurs de charge. Il est ainsi propose d’utiliser la mesure du temps d’extinction duplasma comme un diagnostic complementaire pour les plasmas poussiereux. La comparaison avecdes donnees experimentales existantes [21] soutient cette idee.Comme les poudres se chargent durant la phase plasma, cette charge electrique doit disparaıtredurant la phase post-decharge. Cependant, il a ete observe que les poudres gardent une faiblecharge residuelle et que celle-ci peut etre positive ou negative. Le temps de chargement despoudres etant extremement long a la fin de la phase post-decharge, des fonctions de distributionde charges ont pu etre mesurees pour differentes conditions. Ainsi, la charge moyenne des poudresest negative mais les queues des distributions s’etendent vers les charges positives. L’ecart typedes distributions est aussi plus grand que celui predit theoriquement pour les distributions decharge en phase plasma. Les simulations numeriques montrent que la diffusion des porteurs decharge et plus particulierement la transition de la diffusion ambipolaire vers la diffusion librejoue un role majeur pour expliquer les distributions de charges residuelles observees. Ainsi, unetransition abrupte lorsque la longueur de Debye electronique atteint la longueur de diffusion dureacteur est incapable d’expliquer l’existence de charges residuelles positives. En effet, la vitessede transition est essentielle pour expliquer la charge moyenne et l’ecart type des distributionsmesurees. Des transitions rapides aboutissent a des charges residuelles moyennes plus faibles(en valeurs absolues) et des distributions plus larges.

171

Chapter 7

General conclusion

7.1 General conclusion in English

In this thesis, we focussed on the influence of dust particles on RF discharges from the plasma

ignition to the discharge afterglow. Experimental investigations under different discharge con-

ditions have been performed. Different diagnostics have been used in order to understand the

different processes involved in plasma-dust particle interactions. These allowed us to carefully

analyse the possible effects of dust particles on RF discharges during different times in the life

of the discharge:

1. The first stage of the life of dusty RF discharges is the plasma ignition and the growth

of dust particles. It has been shown that dust particle growth by RF sputtering can be

followed by monitoring plasma and discharge parameters. Therefore, the self bias voltage

evolution in discharges where dust particles are grown by RF sputtering is shown to be

very similar to its evolution in chemically active dusty discharges. Qualitatively the effect

of particle growth on the self-bias can be seen as a consequence of reduction in electron

density due to negative charge accumulation on the dust particles. A reduced self-bias

voltage reduces the ion flux and allows a longer period of electron flow so that a new

equilibrium is established. Collections of dust particles and measurements of their size

for different discharge running times, corresponding to different times during the self bias

voltage evolution pattern, showed that the dust particles increase in size as a function of

time. However strong changes in the self bias voltage do not seem to be correlated to

strong modifications of the particles sizes. The effect of the dust particle growth is also

clearly shown from the evolution of the RF current amplitude which decreases with time

confirming the reduction of electron density due to accumulation of negative charges on

the dust particles.

In order to estimate how much the electrons are affected by growing dust particles, Lang-

muir probe measurements and spectroscopic measurements were performed. They showed

that the EEDF is not Maxwellian and that effective electron temperature increases dur-

ing dust particle growth process. Spectroscopic measurements suggest a depletion of the

172

7.1 CHAPTER 7. GENERAL CONCLUSION

energetic electron tail.

The influence of plasma conditions has also been studied. It has been shown that high

pressures are favourable to the dust particle growth in flowing and static gas. The gas

purity also plays an essential role in the dust particle growth process. It has indeed been

reported that very low base pressure and long pumping time in between experiments are

necessary to achieve high dust particle density and fast dust growth kinetics. The influ-

ence of the input power or RF peak to peak voltage has also been investigated. It has

been found that high power or high RF peak to peak voltage tend to favour dust particle

growth.

Finally it has been observed that the dust particle growth can trigger instabilities in the

plasma: the dust particle growth instabilities (DPGI). It has been found that these insta-

bilities followed a well defined pattern separated into different phases with characteristic

frequencies and amplitudes. Some of these phases have been related to the filamentary

mode and the great void mode previously reported in experiments where the dust particles

were grown by RF sputtering [74]. It has also been found that the growth kinetics have an

influence on the DPGI. Slow growth kinetics lead to DPGI with lower frequencies which

evolve over longer periods of time than for fast growth kinetics.

2. The second stage of the dusty RF discharge life corresponds to the existence of a well

established dense cloud of dust particles. The dust cloud very often exhibits a centimetre

size dust free region with very sharp boundaries in the centre of the plasma called the dust

“void”. It is generally admitted that the dust void is the result of an equilibrium between

an inward electric force that confines the dust particles inside the plasma bulk and an

outward ion drag force which tends to repel the dust particles from the plasma. Under

specific conditions, the dust void can exhibit some instabilities which can be identified

from the electric parameters of the discharge, the plasma glow, and the dust particle cloud

motion. Two kinds of instabilities have been reported.

The first kind of instability is a self excited instability of the dust void consisting of

contraction-expansion sequences (CES) of its size, called the “heartbeat instability”. This

instability was also observed under microgravity conditions with injected dust particles.

This instability exhibits two extreme cases: a low repetition rate case which consists of

fast CES events separated by long periods of time in which the dust void can reach its

equilibrium size, and a high repetition rate case which consists of slower CES events which

follow each other without re-establishment of the equilibrium condition. In both cases,

the void contraction is accompanied by a strong increase of the plasma glow inside the

void region (identified also by a strong increase of the discharge current) which suggests

an ionisation increase that can break the equilibrium between the electric force and the

ion drag force. Between these two extreme cases, the instability exhibits failed or aborted

CES events between the full sequence. It is observed that the more failed CES there are

in between full sequences, the faster the complete CES.

The second kind of instability is due to the growth of a new generation of dust particles

173


inside the void region. Continuous growth of dust particles inside the plasma bulk can

occur if the discharge parameters are suitable, and many generations of dust particles can

follow each other. A new generation of dust particles grows inside the void region of an

older generation and, under specific conditions not yet well understood, this can trigger

an instability of the “void” which stops when the new dust particles are big enough:

the “delivery instability”. This instability is characterised by low frequency and large

amplitude expansion-contraction sequences of the dust void size. The time evolution of

the frequency and the amplitude of the instability is directly linked to the growth kinetics

of new dust particles. This instability sometimes evolves into another one where the dust

void is rotating around the axis of symmetry of the discharge.

3. The last stage is the discharge afterglow. The charge carriers of the plasma diffuse onto

the walls of the chamber and recombine into neutral gas. In a complex plasma, the

charged dust particles play a significant role in the plasma decay. They act as absorption-

recombination sinks and thus drastically shorten the plasma decay time. This has been

very well observed by comparison of measurements of electron density decay in dust-free

afterglow plasma and dusty afterglow plasma. It has been shown that the dust particle

absorption frequency does not depend linearly on the dust particle density. This is due to

strong dust-dust interactions which increase with the dust density and reduce their mean

charge and thus the fluxes of charge carriers on dust particles. It is proposed to use the

plasma decay time measurement as a complementary diagnostic for complex plasma.

As the dust particles are electrically charged, their charges must also relax during the

afterglow phase. However, it has been shown experimentally that the dust particles do

keep a small residual electric charge which can be either positive or negative. As the dust

particle charge time becomes very long in the late discharge afterglow, it has been possible

to measure the residual dust charge distribution. It has thus been shown that the mean

residual charge is negative but the tail of the distribution extends in the positive charge

region. It has also been found that the standard deviation of the dust charge distribution in

the afterglow plasma is larger than the one predicted for a running discharge. Simulations

have shown that the diffusion process for charge carriers and especially the transition

from ambipolar-to-free diffusion is essential in explaining the experimental dust charge

distributions. Indeed an abrupt transition from ambipolar-to-free diffusion, when the

electron Debye length reaches the diffusion length, is not able to explain the existence of

positively charged dust particles in the late afterglow. It has also been shown that the

speed of the transition is essential to explain the mean residual charge and the width of the

distribution. A fast transition leads to smaller mean residual charge and larger standard

deviation of the dust charge distribution.

To conclude, a detailed analysis of different phenomena that can occur in RF dusty discharges

has been performed. The work presented in this thesis has led us to a better understanding of

complex plasmas at every stage of the life of the discharge.

174


7.2 Conclusion générale en français

Dans cette these, l’influence de la presence de poudres sur des decharges RF depuis leurallumage jusqu’a leur extinction a ete etudiee. Des experiences sous differentes conditions ontete realisees. Differents diagnostics ont ete utilises afin de comprendre les differents processusimpliques dans les interactions plasma-poussiere. Cela nous a permis d’analyser finement lesdifferents phenomenes pouvant affecter une decharge RF poussiereuse aux differents instants deson evolution.

– La premiere etape dans la vie d’une decharge RF poussiereuse est l’allumage du plasma et lacroissance des poudres. Il a ete demontre que la formation des poussieres par pulverisationpeut etre suivie a l’aide des parametres de la decharge et du plasma. Ainsi, l’evolution dela tension d’autopolarisation est tres similaire a celle des decharges ou les poudres sontformees dans des plasmas reactifs : qualitativement, l’effet de croissance des poussieressur la tension d’autopolarisation peut etre vu comme une consequence de la diminutionde la densite electronique due a l’accumulation de charges negatives sur les poudres. Unereduction de la tension d’autopolarisation induit une reduction du flux d’ions positifs etun allongement de la duree du flux d’electrons necessaires au nouvel equilibre. La collectedes poudres et la mesure de leurs tailles pour differentes durees du plasma ont montre queles poudres grossissent avec le temps mais que les fortes modifications de la tension d’au-topolarisation ne sont pas accompagnees d’une forte variation de la taille des poussieres.La croissance des poudres peut aussi etre suivie grace a l’evolution de l’amplitude de l’har-monique fondamentale du courant RF. En effet, celle-ci diminue durant la formation despoussieres confirmant la reduction de la densite electronique due a l’accumulation descharges negatives sur les poudres.Afin d’estimer l’effet des poudres sur la fonction de distribution en energie des electrons,des mesures de sonde de Langmuir associees a de la spectroscopie d’emission ont ete ef-fectuees. Il a ainsi ete demontre que la fonction de distribution en energie des electronsn’est pas maxwellienne et que la temperature electronique effective croıt durant la forma-tion des poudres. La spectroscopie d’emission semble indiquer que la queue energetique dela fonction de distribution se depeuple dans le meme temps.L’influence des differents parametres de la decharge sur la croissance des poudres a aussiete etudiee. Il a ainsi ete montre que les pressions elevees sont favorables a la formationde poussieres que ce soit avec ou sans flux de gaz. La purete du gaz joue aussi un roleessentiel dans le processus de croissance. En effet, des pressions de base tres basses ainsique de longues periodes de pompage entre chaque experience sont necessaires pour obtenirdes densites de poudres elevees et des cinetiques de croissance rapides. L’influence de lapuissance injectee ou de la tension RF a aussi ete etudiee. Il apparaıt que de fortes puis-sances ou des tensions RF elevees favorisent la formation des poussieres.Enfin, la croissance de poudres peut declencher des instabilites du plasma : les insta-bilites de croissance des poudres. Ces instabilites suivent un schema d’evolution precisconstitue de differentes phases de frequences et d’amplitudes definies. Certaines de cesphases peuvent etre comparees aux modes filamentaire et “great void” precedemment ob-serves durant des experiences ou les poudres etaient formees par pulverisation RF [74].La cinetique de croissance des poudres a une influence sur les instabilites : des cinetiqueslentes induisent des instabilites basses frequences qui evoluent sur des periodes de tempsbien plus longues que dans le cas de cinetiques rapides.

– La seconde etape de la vie d’une decharge RF poussiereuse coıncide avec l’existence d’unnuage dense de poudres piege dans le plasma. Une region sans poudre, d’environ 1 cmde diametre et aux contours clairement delimites, est souvent presente dans le nuage aucentre du plasma. Cette region est appelee « void ». Il est communement admis que ce« void » est le resultat d’un equilibre entre une force electrostatique confinante et la forcede friction des ions qui a tendance a expulser les poudres du plasma. Sous certaines condi-tions, ce « void »devient instable. Ces instabilites peuvent etre reperees sur les parametreselectriques de la decharge, sur la luminosite du plasma et sur la dynamique du nuage du

175


poudres. Deux types d’instabilites ont ete observes :Le premier type est une instabilite auto-excitee du « void » appelee l’instabilite « heart-

beat ». Elle se compose de battements, c’est a dire de sequences de contraction-expansion(SCE) des dimensions du « void ». Ce type d’instabilite a egalement ete observe lorsd’experiences en microgravite avec des poudres micrometriques injectees. Le « heartbeat »

presente deux cas extremes : un cas avec un taux de repetition faible qui se compose de SCEtres rapides separees par de longues periodes durant lesquelles le systeme a le temps de re-tourner a l’equilibre, et un cas avec un fort taux de repetition compose de SCE plus lentesmais qui se succedent rapidement sans retour a l’equilibre. Dans les deux cas la contrac-tion s’accompagne d’une forte augmentation de la luminosite du plasma a l’interieur du« void » (aussi associee a une forte augmentation du courant) suggerant un accroissementde l’ionisation rompant l’equilibre entre la force electrostatique et la force de friction desions. Entre ces deux cas extremes, l’instabilite presente des SCE avortees entre les SCEcompletes. Plus le nombre de SCE avortees est grand entre chaque SCE complete, plus cesdernieres sont rapides.

Le second type d’instabilite est amorce par la croissance d’une nouvelle generation depoudres dans le « void ». Lorsque les parametres de la decharge sont favorables, il peut yavoir une formation continue de poussieres dans le plasma et de nombreuses generationspeuvent ainsi se succeder. Lorsqu’une nouvelle generation de poudres croıt dans le « void »

de l’ancienne generation, elle peut, sous certaines conditions (non-determinees pour l’ins-tant), declencher une instabilite du « void » qui ne s’arrete que lorsque les poudres de lanouvelle generation ont atteint une taille et/ou une densite critiques. Cette instabilite estmarquee par des sequences d’expansion-contraction des dimensions du « void » a bassefrequence et de grande amplitude. L’evolution temporelle de la frequence et de l’amplitudedes oscillations est directement reliee a la croissance de la nouvelle generation de poudres.Cette instabilite se transforme parfois en une rotation du void autour de l’axe de symetriede la decharge.

– La derniere etape est la phase post-decharge. Les porteurs de charge du plasma diffusent etse recombinent en gaz neutre. Dans un plasma poussiereux, les poudres chargees jouent unrole significatif dans la disparition du plasma. Elles agissent comme des pieges sur lesquelsles porteurs de charge se recombinent et ainsi reduisent fortement le temps d’extinctiondu plasma. Cet effet a ete mis en evidence experimentalement en comparant les tempsd’extinction dans des decharges avec ou sans poudres. Il a ete demontre que la frequenced’absorption des porteurs de charge par les poudres n’augmente pas lineairement avec ladensite de poussieres. Ceci est du au fait que les interactions poudre-poudre, augmentantavec la densite de poudres, reduisent la charge electrique de ces dernieres et donc les fluxde porteurs de charge. Il a ainsi ete propose d’utiliser la mesure du temps d’extinction duplasma comme un diagnostic complementaire pour les plasmas poussiereux.Comme les poudres se chargent durant la phase plasma, cette charge electrique doit doncdisparaıtre durant la phase post-decharge. Cependant, il a ete observe que les poudresgardent une faible charge residuelle et que celle-ci peut etre positive ou negative. Le tempsde chargement des poudres etant extremement long a la fin de la phase post-decharge, desfonctions de distribution de charges ont pu etre mesurees pour differentes conditions. Ainsi,la charge moyenne des poudres est negative mais les queues des distributions s’etendentvers les charges positives. L’ecart type des distributions est aussi plus grand que celuipredit theoriquement pour les distributions de charge en phase plasma. Les simulationsnumeriques montrent que la diffusion des porteurs de charge et plus particulierement latransition de la diffusion ambipolaire vers la diffusion libre joue un role majeur pour expli-quer les distributions de charges residuelles observees. Ainsi, une transition abrupte lorsquela longueur de Debye electronique atteint la longueur de diffusion du reacteur est incapabled’expliquer l’existence de charges residuelles positives. En effet, la vitesse de transition estessentielle pour expliquer la charge moyenne et l’ecart type des distributions mesurees. Destransitions rapides aboutissent a des charges residuelles moyennes plus faibles (en valeurabsolue) et des distributions plus larges.

176


En conclusion, une analyse detaillee des differents phenomenes pouvant survenir dans desdecharges RF poussiereuses a ete realise. Les travaux presentes dans cette these nous ont permisde mieux comprendre les differents processus physiques impliques dans les interactions plasma-poudre.

177

Appendix A

Dust charge distribution in running

discharges

The Eq.2.18 can be solved this way. Taking x = Qd−Qmean and τ = t/τQ and then introducing

the dimensionless variable ξ = x/σ, Eq.2.18 can be rewritten as:

∂W

∂τ=

( ∂2

∂ξ2+

∂

∂ξξ)W (A.1)

In the following, the initial condition is:

W (Qd, 0) = δ(Qd − Q0) (A.2)

with Q0 a value of Qd within the linear range of the current. This condition can be rewritten

as:

W (ξ, 0) = δ(ξ − ξ0) (A.3)

As W is a probability we have:

∫ +∞

−∞

dQdW (Qd, t) =

∫ +∞

−∞

dξW (ξ, t) = 1 (A.4)

We introduce the time dependant position variable y = ξeτ . We thus have :

∂

∂ξ= eτ ∂

∂y(A.5)

and thus

∂

∂τW (y(τ), τ) =

∂

∂τW (y(τ), τ) +

∂y

∂τ

∂

∂yW (y(τ), τ) (A.6)

∂

∂τW (y(τ), τ) =

∂

∂τW (y(τ), τ) + y

∂

∂yW (y(τ), τ) (A.7)

178

A.0 APPENDIX A. DUST CHARGE DISTRIBUTION IN RUNNING DISCHARGES

Eq.A.1 can be rewritten as

∂

∂τW (y(τ), τ) = e2τ ∂2

∂y2W (y(τ), τ) + W (y(τ), τ) (A.8)

We take W = eτf , Eq.A.8 becomes

∂

∂τf = e2τ ∂2

∂y2f (A.9)

which is the equation of diffusion with a time-dependant diffusion coefficient D(τ) = e2τ . The

solution is [177]:

f(y, τ) =(4π

∫ τ

0dτ ′D(τ ′)

)−1/2exp

(− (y − y0)

2

4∫ τ0 dτ ′D(τ ′)

)(A.10)

which gives [91]:

W (Qmean + x, t) =1

σ√

2π(1 − e−2t/τQ)exp

(− (x − x0e

−t/τQ)2

2σ2(1 − e−2t/τQ)

)(A.11)

From Eq.A.11, it is clear that the charge distribution is Gaussian at all the times and approaches

with a time constant τQ a steady states with a mean dust particle charge Qmean and a variance

σ which depend on the charging currents. From Eq.2.19 and Eq.2.20, the variance σ can be

rewritten as: (σ

e

)2=

τQ

2τc(A.12)

where τC = e/(Ii − Ie). It has been shown that τQ represents the time for charge fluctuations

and τc represents the mean time for collisions between a dust particles and a charge species [91].

179

Appendix B

Empirical Mode Decomposition

(EMD) and Hilbert-Huang (HH)

spectrum

Data analysis is a procedure which allows us to understand the physical processes implicated in

observed phenomena. However, problems are often encountered when one tries to analyse data:

• the total data span is too short

• the process which generates the data is non-linear

• the data are non stationary

The most common way to analyse data is Fourier analysis, but this is very limited for analysis

of non-linear and/or non-stationary data. Indeed Fourier analysis supposes that the data are

stationary and that the signal may be decomposed using trigonometric functions with fixed

amplitudes over the whole span of the data. Consequently, local time variations of a signal are

very hard to track. Moreover, non-linear data need to be decomposed over a wide range of

harmonics and as a result, the energy will be spread over a large frequency range.

The characteristics of the function used to analyse a signal can be misinterpreted and taken

as characteristic of the signal itself. Consequently, it is better to choose analysis functions in

accordance with the intrinsic structure of the analysed signal. For this purpose, Huang et al.

introduced a new method to analyse non-linear and non- stationary time series [178]. It is based

on the empirical mode decomposition (EMD) and the Hilbert-Huang transform (HHT). The

key aspect of this method is to decompose the signal into a finite number of intrinsic mode

functions (IMF) based on local characteristic time scales of the data and then to determine the

instantaneous frequency of these IMFs using the HHT.

180

B.1 APPENDIX B. EMD AND HH SPECTRUM

B.1 The empirical mode decomposition

The main idea in EMD is to consider oscillations in the signal at a very local level. It is needed

to identify the time scale that will reveal the physical characteristics of the studied process.

Then, these time scales have to be extracted into intrinsic mode functions (IMF). Consequently,

the EMD is a data sifting process which eliminates locally riding waves as well as the local

asymmetry in the time series profile [179]. For a signal X(t), the effective algorithm for EMD

can be summarized as [178]:

1. Identify all extrema of X(t)

2. Interpolate between maxima using cubic spline line to obtain the envelope emax(t) and

interpolate between minima to obtain the envelope emin(t)

3. Calculate the mean m(t) = (emax(t) − emin(t))/2

4. Extract the first component h1(t) = X(t) − m(t)

5. Iterate on m(t)

The step 1 to 4 may have to be repeated several times until the component h(t) can be considered

as zero mean according to some stopping criterion[178]: each hi(t) is treated as the signal and

when it fits the stopping criterion, it is considered as the IMF and m(t) is treated as the signal.

At the end of the process, the signal X(t) is decomposed into a finite number n of IMFs Cj(t)

and a residue rn(t). The original signal can be reconstructed by superimposition of the IMFs

and the residue as follows:

X(t) = rn(t) +n∑

j=1

Cj(t) (B.1)

By definition an IMF has to satisfy two conditions [178]:

• The number of extrema must be equal or differ at most by one from the number of zero

crossings. It is similar to the narrow band requirements for a stationary Gaussian process

• At any point, the mean value of the envelope defined by the local maxima and the envelope

defined by the local minima is zero. It ensures phase function without bias and modifies

the classical global requirement to a local one. This is necessary so that the instantaneous

frequency will not have unwanted fluctuations induced by asymmetric wave forms.

The stopping criterion is thus very important in order to obtain physically meaningful IMFs.

The sifting process is terminated when two conditions are fulfilled:

1. The number of extrema must differ at most by one from the number of zero crossings.

2. The mean between the upper and lower envelope must be close to 0 according to some

criterion

181


The stopping criterion is based on two thresholds θ1 and θ2 [180]. It aims at guaranteeing

globally small fluctuations while taking into account locally large excursions. It is based on the

mode amplitude a(t) = (emax(t) − emin(t))/2 and the evaluation function σ(t) = |m(t)/a(t)| so

that the sifting is iterated until σ(t) < θ1 for some prescribed fraction of the signal (1−α) while

σ(t) < θ2 for the remaining fraction. The set of values used in this thesis is α = 0.01 θ1 = 0.05

and θ2 = 0.5.

B.2 The Hilbert Huang spectrum

After the signal X(t) has been decomposed into IMF, the Hilbert-Huang transform (HHT) can

be applied. The HHT on an IMF Cj(t) is defined as:

Cj(t) =1

πP

∫ +∞

−∞

Cj(t′)

t − t′dt′ (B.2)

where P indicates the Cauchy principal value. It is then possible to deduce the amplitude aj ,

the phase ϕj and the instantaneous frequency ωj :

aj(t) =√

C2j (t) + C2

j (t) (B.3)

ϕj(t) = arctan( Cj(t)

Cj(t)

)(B.4)

ωj(t) =dϕj(t)

dt(B.5)

The original signal can be expressed as:

X(t) = Ren∑

j=1

aj(t) exp(ı

∫ωj(t)dt

)(B.6)

where Re stands for real part.

By definition IMFs have positive frequencies because they are symmetric with respect to the

local mean and consequently admit a well-behaved HHT 1.

As can be seen from Eq.B.6, IMFs represent a generalized Fourier expansion. The main improve-

ment is that the variable amplitude and the instantaneous frequency enable the expansion to

accommodate non-stationary data. The frequency and amplitude modulations are also clearly

separated.

From Eq.B.6, it is also possible to construct the frequency-time distribution of the amplitude

designated as the Hilbert-Huang amplitude spectrum H(ω, t) (HHS) [178]. The time resolution

in the HHS can be as precise as the sampling rate of the data and the frequency resolution is

arbitrary. The lowest extractable frequency is fmin = 1/T where T is the time duration of the

data and the highest frequency extractable is fmax = 1/(k∆t) where ∆t is the sampling rate of

1It is shown in Ref.[178] that the Hilbert transform can lead to non-physical negative frequencies if the signalis not locally symmetric with respect to the zero mean level.

182


the data and k the minimum number of points necessary to define the frequency accurately. For

a sine wave, k = 5 [178, 179].

From the HHS, it is possible to extract the marginal Hilbert spectrum defined as [178]:

h(ω) =

∫ T

0H(ω, t)dt (B.7)

h2(ω) =

∫ T

0H2(ω, t)dt (B.8)

It represents the cumulative amplitude (or squared amplitude) over the entire data span and

offers a measure of the total energy contribution from each frequency. It is different from the

Fourier spectrum in a sense that in the Fourier spectrum the existence of energy at a frequency

ω means that a sine or cosine wave with this frequency persists during all the data span while

for the marginal Hilbert spectrum it only means that during the whole span of the data such a

wave with frequency ω has a high probability to appear locally.

It is also possible to define the instantaneous energy density level IE as [178]:

IE(t) =

∫

ωH2(ω, t)dω (B.9)

As IE depends on time, it can be used to check the energy fluctuations of the signal.

B.3 Examples

B.3.1 Analysis of an analytic non-linear signal

Let’s considered the following signal X(t) 2:

X(t) = cos(ωt + ǫ sinωt

)(B.10)

where ω = 125.6 Hz (frequency f = 20 Hz) and ǫ = 0.1 (Fig.B.1). This is a non-linear wave with

a small frequency modulation. The function is solution of the following non-linear equation:

d2x

dt2+ (ω + ǫω cos ωt)2x − ǫω2 sinωt(1 − x2)1/2 = 0 (B.11)

As X(t) is already an IMF, the Fourier spectrogram and the HHS are presented in Fig.B.2.

As it can be seen, the Fourier analysis enlightens the main frequency at 20 Hz but also shows

harmonics which are necessary to reproduce the non-linear signal. On the other hand, the HHS

shows only one frequency around 20 Hz which is modulated at a frequency of 20 Hz.

The harmonic content obtained from the Fourier analysis arises only from the decomposition of

2A more detailed description of this example with parameter values is given in Ref.[178]

183


0 0.1 0.2 0.3 0.4 0.5

−1

−0.5

0

0.5

1

Time (s)

a.u

.

SignalPure cos. wave

(a) Signal

0 0.02 0.04 0.06

−1

−0.5

0

0.5

1

Time (s)

a.u

.

SignalPure cos. wave

(b) zoom of a)

Figure B.1: Non-linear signal of a wave with small frequency modulation and pure cosine wavewith the same mean frequency

Fourier spectrogram

Time (s)

Fre

qu

ency

(H

z)

0 0.1 0.2 0.3 0.4 0.5

20

40

60

80

−10

−5

0

5

(a) Fourier spectrogram

Time(s)

Fre

qu

ency

(H

z)

Hilbert−Huang spectrum

0 0.1 0.2 0.3 0.4 0.5

20

40

60

80

−5

−4

−3

−2

−1

0

(b) Hilbert-Huang spectrum

Figure B.2: Fourier spectrogram and Hilbert-Huang spectrum of a non-linear wave signal withsmall frequency modulation

184


a non-linear wave as a sum of linear waves. Indeed Eq.B.10 can be written as:

X(t) = cos(ωt + ǫ sinωt

)

= cos ωt cos(ǫ sinωt) − sinωt sin(ǫ sinωt)

≃ cos ωt − ǫ sin2 ωt ≃ (1 − 1

2ǫ) cos ωt +

1

2ǫ cos 2ωt (B.12)

Consequently this example shows that the harmonic components of a signal can be the result

0 0.05 0.1 0.15 0.218

19

20

21

22

Time (s)

Fre

qu

ency

(H

z)

From Hilbert transform

from theory

Figure B.3: Evolution of the frequency of the non-linear wave

of using a linear system to simulate a non-linear one.

From Eq.B.10, it is also possible to obtain frequency modulation from classical wave theory:

Ω =dθ

dt= ω(1 + ǫ cos ωt) (B.13)

In Fig.B.3, the frequency obtained from the Hilbert transform is compared to the theoretical

one obtained from Eq.B.13. As can be seen, it is in very good agreement. It thus confirms that

a deformed sinusoidal wave can be interpreted as an intra-wave frequency modulated wave.

B.3.2 EMD of a heartbeat signal

The empirical mode decomposition has been applied to a heartbeat signal (AC current; see

Chap.5). The results are presented in Fig.B.4. As can be seen, the signal is composed of a finite

number of modes.

The modes with the highest amplitude (IMF 5 to 7) are the main modes of the signal. Indeed

as can be seen in Fig.B.5, the sum of these modes reproduce the main characteristics of the

measured signal.

The Hilbert transform has been applied to the IMFs. The HHS and marginal spectrum have

been plotted (Fig.B.6). As can be seen in Fig.B.6(a), the signal has a main frequency centred

around 20 Hz and exhibits a slightly weaker component with large frequency excursion centred

around 40 Hz.

185


−1

0

1

Sig

nal

Time (s)

IMF

1

Time (s)

−1

0

1

IMF

2

Time (s)

IMF

3

Time (s)

−1

0

1

IMF

4

Time (s)IM

F5

Time (s)

−1

0

1

IMF

6

Time (s)

IMF

7

Time (s)

−1

0

1

IMF

8

Time (s)

IMF

9

Time (s)

0 0.2 0.4−1

0

1

IMF

10

Time (s)0 0.2 0.4

R

Time (s)

Figure B.4: Heartbeat signal (AC current) and its IMFs from EMD

0.150.075 0.225−1

−0.5

0

0.5

1

Time (s)

a.u

.

Signal∑n=7

n=5IMFn

Figure B.5: Heartbeat signal (solid line) and the sum of its largest amplitude IMFs (dashed-line)

186


The marginal Hilbert spectrum is also compared to the Fourier spectrum of the signal (Fig.B.6(b)).

The two frequencies previously observed are present in both spectra. However, in the marginal

Hilbert Spectrum they are not represented by sharp peaks indicating that the frequencies are

effectively varying around their mean values while in the Fourier spectrum those frequencies are

observed with sharp peaks and accompanied by additional harmonics necessary to reconstruct

the non-linear data. This suggests the additional harmonics are not characteristic of the physical

process leading to the non-linear signal, but merely provide a linear reconstruction.

Time(s)

Fre

qu

ency

(H

z)

0 0.1 0.2 0.3 0.4 0.5

20

40

60

80

100

−5

−4

−3

−2

−1

0

(a) Hilbert-Huang spectrum

20 40 60 80 100

10−4

10−2

100

Frequency (Hz)

a.u

.

marginal Hilbert spectrum

Fourier spectrum

(b) marginal Hilbert Spectrum and Fourier spec-trum

Figure B.6: HHS and marginal Hilbert spectrum of the heartbeat signal

187

Appendix C

Matlab program for dust resisual

charge simulation

C.1 Main program

%electrons and ions density are computed idependantly%dust charge computed through a Focker Planck algorithmfunction [ni,ne,CR,CRevol,t,tde,tdi,tabs,Te]=ModelResCharge3(ts,nd,n0)

%ts: simulation time%n0: initial density%nd: dust densityttinfty=0.900e-3; %electron temperature relaxation time (to modify as a function of studied conditions)te0=100; %initial Te/Ti)tdinfty=3.5e-3; %diffusion time (to modify as a function of studied conditions)ldi0=18e-6; %initial debye length for ni=5e9 cm-3lambda=1e-2; %reactor diffusion length%time step and number of stepdt=1e-6; %time stepN=ts/dt; %numner of step%Number of dust particles

Np=500;rd=190e-9;%dust radiusVeq=Np/nd; %equivalent volume of the Nd dust particles

%initialisation of parameterst=zeros(1,N);Te=zeros(1,N);td=zeros(1,N);tabs=zeros(1,N);tde=zeros(1,N);tdi=zeros(1,N);ni=zeros(1,N);ne=zeros(1,N);CRevol=zeros(int16(N/10)+1,Np);

% %initialisation de la charge sur les poudres[CR,ne(1),ni(1)]=PlasmaActif(Np,nd,n0,rd,10e-7,te0,-700);disp(’*************’);disp(’initialisation finished’);disp(’*************’);

wx=1; %time step to save charge distributionfor i=1:N

if (mod(i,100)==0)

188

C.1 APPENDIX C. PROGRAM

disp(’still running’);disp(i);

endTe(i)=CalculerTe(dt*(i-1),ttinfty,te0);%density of electrons and ionsif i =1

nev=ne(i-1);niv=ni(i-1);

elsenev=ne(1);niv=ni(1);

endniabs=0;neabs=0;tabs(i)=0;for j=1:Np

temps=0;tm1=0; %charging time until now for the jth dust particlek=0;while temps<dt

%calcul des courants d’ions et d’electrons%Calcul du potentiel de surface de la poudre traiteePhiS=CR(j)*1.602e-19/(4*pi*8.854e-12*rd);if nev<0;

nev=0;disp(’ne neg’);disp(i);return;%break;

endif niv<0;

niv=0;disp(’ni neg’)disp(i);return;%break;

endI0i=4*pi*rd*rd*niv*1.602e-19*sqrt(1.38e-23*(1.602e-19*0.03/1.38e-23)

/(2*pi*6.63e-26))*(1+30e-3*Te(i));I0e=-4*pi*rd*rd*nev*1.602e-19*sqrt(1.38e-23*(1.602e-19*0.03/1.38e-23)*Te(i)

/(2*pi*9.11e-31));%ion currentif PhiS<0

Ii=I0i*(1-PhiS/(0.03));else

Ii=I0i*exp(-PhiS/(0.03));end%electron currentif PhiS>0

Ie=I0e*(1+PhiS/(Te(i)*0.03));else

Ie=I0e*exp(PhiS/(Te(i)*0.03));end%proba collection electronpre=-Ie/1.602e-19;%proba collection ionpri=Ii/1.602e-19;ptot=pre+pri;%charging time stepR1=rand(1);while (R1==1 — R1==0)

R1=rand(1);enddeltat=-log(1-R1)/ptot;if k==0

189


tabs(i)=tabs(i)+deltat;endtm1=temps;temps=temps+deltat;if temps<dt

%choice of the absorbed particlesR2=rand(1);if R2<(pre/ptot)

CR(j)=CR(j)-1; %electronsneabs=neabs+1;

elseCR(j)=CR(j)+1;%ionsniabs=niabs+1;

endendif temps>dt

R3=rand(1);P=1-exp(-(dt-tm1)*ptot);if R3<P

R2=rand(1);if R2<(pre/ptot)

CR(j)=CR(j)-1; %electronsneabs=neabs+1;

elseCR(j)=CR(j)+1;%ionsniabs=niabs+1;

endend

endif deltat<0

disp(’crash);disp(temps);disp(deltat);return;

endk=k+1;if k>1000000

disp(’crash2’);disp(i);disp(deltat);return;

endend

endt(i)=(i-1)*dt;tabs(i)=tabs(i)/Np;if i==1

td(i)=Calcultd(t,tdinfty,ttinfty,Te(i));ldi=sqrt(8.85e-12*0.03/(ni(1)*1.602e-19));lde=sqrt(8.85e-12*Te(1)*0.03/(ne(1)*1.602e-19));

elseldi=sqrt(8.85e-12*0.03/(ni(i-1)*1.602e-19)); %longuer de debye ioniqueif ne(i-1)>0

lde=sqrt(8.85e-12*Te(i)*0.03/(ne(i-1)*1.602e-19));else

lde=10000; %artefact for ne=0 w/o numerical pbendld=lde*ldi/sqrt(lde^2 +ldi^2);[corr De,corr Di]=CorrectionDiffusion2(lambda,lde);%Correction to ambipolar diffusiontd(i)=Calcultd(t,tdinfty,ttinfty,Te(i)); %diffusion time%if td(i)<0

disp(’td neg’);return;

190


end%correction diffusion times%Phys Rev A 7,781 (1972)tde(i)=td(i)./corr De;tdi(i)=td(i)./corr Di;%evolution of ion and electrondensitiesne(i)=ne(i-1)-dt*ne(i-1)*(1/(tde(i)*1))-neabs/Veq;ni(i)=ni(i-1)-dt*ni(i-1)*(1/(tdi(i)*1))-niabs/Veq;if ne(i)<0

ne(i)=0.;end

endif ( i==1 — mod(i,10)==0)

CRevol(wx,:)=CR(:);wx=wx+1;

end%stop when charging time biger than diffusion timeif (tabs(i)>20.*tdinfty)

disp(’i fin’);disp(i);disp(ldi);return;

endend

C.2 Subprogram

C.2.1 Initialisation of dust particle charges

function [CR,ne,ni]=PlasmaActif(Np,nd,n0,rd,dt,Te,c0)

%charge on dust taking ito account quasi neutrality and dust density%some parameters such as dt and the starting dust charge must be adjust manually to improve speed%land stability

Veq=Np/nd; %volume occupe par les 500 poudresni=n0;ne=n0;neprev=ne+0.5*ne;Zdmeanprev=5000;Zdvarprev=5000;diff=100.*abs(neprev-ne)/ne;diff2=100;diff3=100;%dt=1e-6;k=1;%Te=100;CR=c0*ones(1,Np);while (diff>0.05 | diff2>0.05 | diff3>0.05 | k<25)

i=1;disp(k);Zdmeanprev=mean(CR);Zdvarprev=std(CR); for j=1:Np

%disp(j);temps=0;while temps<dt

%disp(temps);%calcul des courants d’ions et d’electrons%Calcul du potentiel de surface de la poudre traiteePhiS=CR(j)*1.602e-19/(4*pi*8.854e-12*rd);I0i=4*pi*rd*rd*ni*1.602e-19*sqrt(1.38e-23*(1.602e-19*0.03/1.38e-23)

/(2*pi*6.63e-26));

191


I0e=-4*pi*rd*rd*ne*1.602e-19*sqrt(1.38e-23*(1.602e-19*0.03/1.38e-23)*Te/(2*pi*9.11e-31));

%courant d’ionif PhiS<0

Ii=I0i*(1-PhiS/(0.03)); %car Te(j)=Te/Ti en realiteelse

Ii=I0i*exp(-PhiS/(0.03));end%courant d’electronsif PhiS>0

Ie=I0e*(1+PhiS/(Te*0.03));else

Ie=I0e*exp(PhiS/(Te*0.03));end%proba collection electronpre=-Ie/1.602e-19;%proba collection ionpri=Ii/1.602e-19;ptot=pre+pri;%calcul pas tempsR1=rand(1);if (R1==1 — R1==0)

R1=rand(1);enddeltat=-log(1-R1)/ptot;temps=temps+deltat;if temps<dt

%choix particule absorbeeR2=rand(1);if R2<(pre/ptot)

CR(j)=CR(j)-1;else

CR(j)=CR(j)+1;end

endif deltat<0

disp(’crash’);disp(temps);return;

endend

endendneprev=ne;ne=ni+sum(CR)/Veq;diff=100.*abs(neprev-ne)/ne;diff2=100.*abs(Zdmeanprev-mean(CR))/mean(CR);diff3=100.*abs(Zdvarprev-std(CR))/std(CR);k=k+1;

endfigure(5);clf;hist(CR,20);xlabel(’Charge (e)’,’fontsize’,16);ylabel(’Number of dust particles’,’fontsize’,16);

C.2.2 Characteristic times and coefficients

function [Te]=CalculerTe(t,tautinfty,te0)

%fonction pour Calculer la temperature electronique (Te/Ti) dans l’afterglow%d’un dusty plasma%On a resolu analitiquement l’equation 3 de%Phys. Rev. Lett. 90(5), 055003 (2003)

192


fte0=abs((sqrt(te0)-1)/(sqrt(te0)+1));Te=((fte0*exp(-t/tautinfty)+1)/(1-fte0*exp(-t/tautinfty)))^2;

function [td]= Calcultd(t,tdinfty,ttinfty,Te)

%Calcul pour calculer le temps caracteristique de%perte du plasma par diffusion%d’apres Phys. Rev. Lett. 90(5), 055003 (2003)

%Te=CalculerTe(t,ttinfty,te0);td=1./(0.5*(1+Te)/tdinfty);

function [corr De,corr Di]=CorrectionDiffusion(lambda,lde)

%program pour calculer les corrections a la diffusion ambipolaire suivant%les donnees tirees de l’article Phys Rev A 7, 781 (1973)

%rapport de la longueur de diffusion sur la longueur de Debye electroniqueratio2=(lambda/lde)^2;%calcul des coefficeints de correctionif ratio2<= 1e-1

pente1=(9.-8.)/(7e-2 - 1e-1);b1=9-pente1*7e-2;corr De=(pente1*ratio2) + b1;if corr De>100;

corr De=100;endcorr Di=0.5;

endif ( ratio2>1e-1 & ratio2<=5e-1)

pente1=(8-5.5)/(1e-1 - 5e-1);b1=5.5-pente1*5e-1;corr De=(pente1*ratio2) + b1;%pente2=(5e-1 - 6e-1)/(1e-1 - 5e-1);b2=6e-1 -pente2*5e-1;corr Di=(pente2*ratio2) + b2;

endif ( ratio2>5e-1 & ratio2<=1)

pente1=(5.5-4.8)/(5e-1 - 1.);b1=4.8-pente1*1;corr De=(pente1*ratio2) + b1;%pente2=(6e-1 - 7e-1)/(5e-1 - 1.);b2=7e-1 -pente2*1;corr Di=(pente2*ratio2) + b2;

endif ( ratio2>1 & ratio2<=5)

pente1=(4.8-2.9)/(1 - 5);b1=2.9-pente1*5;corr De=(pente1*ratio2) + b1;%pente2=(7e-1 - 1.1)/(1 - 5);b2=1.1-pente2*5;corr Di=(pente2*ratio2) + b2;


pente1=(2.9-2.4)/(5-10);b1=2.4-pente1*10;corr De=(pente1*ratio2) + b1;%pente2=(1.1-1.2)/(5-10);b2=1.2-pente2*10;

193


corr Di=(pente2*ratio2) + b2;endif ( ratio2>10 & ratio2<=50)

pente1=(2.4-1.3)/(10 - 50);b1=1.3-pente1*50;corr De=(pente1*ratio2) + b1;%pente2=(1.2-1.25)/(10 - 50);b2=1.25-pente2*50;corr Di=(pente2*ratio2) + b2;





endif ( ratio2>500 & ratio2<=1e3)

pente1=(1.20-1.15)/(500 - 1e3);b1=1.15-pente1*1e3;corr De=(pente1*ratio2) + b1;%pente2=(1.15-1.1)/(500 - 1e3);b2=1.1-pente2*1e3;corr Di=(pente2*ratio2) + b2;

endif ( ratio2>1e3 & ratio2<=5e3)

pente1=(1.15-1.1)/(1e3 - 5e3);b1=1.1-pente1*5e3;corr De=(pente1*ratio2) + b1;%pente2=(1.1-1)/(1e3 - 5e3);b2=1.-pente2*5e3;corr Di=(pente2*ratio2) + b2;


pente1=(1.1-1)/(5e3 - 1e4);b1=1.-pente1*1e4;corr De=(pente1*ratio2) + b1;%pente2=(1.-1.)/(5e3 - 1e4);b2=1.-pente2*1e4;corr Di=(pente2*ratio2) + b2;

endif ratio2>1e4

corr De=1;corr Di=1;

end

function [corr De,corr Di]=CorrectionDiffusion2(lambda,lde)

%program pour calculer les corrections a la diffusion ambipolaire suivant

194


%les donnees tirees de l’article Phys Rev A 7, 781 (1973)%avec resultats de freiberg pour les electrons

%rapport de la longueur de diffusion sur la longueur de Debye electroniqueratio2=(lambda/lde)^2;%calcul des coefficeints de correctionif ratio2<= 1e-1

pente1=(40.-30.)/(7e-2 - 1e-1);b1=40-pente1*7e-2;corr De=(pente1*ratio2) + b1;if corr De>100;

corr De=100;endcorr Di=0.5;

endif ( ratio2>1e-1 & ratio2<=5e-1)

pente1=(30-15)/(1e-1 - 5e-1);b1=15-pente1*5e-1;corr De=(pente1*ratio2) + b1;%pente2=(5e-1 - 6e-1)/(1e-1 - 5e-1);b2=6e-1 -pente2*5e-1;corr Di=(pente2*ratio2) + b2;

endif ( ratio2>5e-1 & ratio2<=1)

pente1=(15-10)/(5e-1 - 1.);b1=10-pente1*1;corr De=(pente1*ratio2) + b1;%pente2=(6e-1 - 7e-1)/(5e-1 - 1.);b2=7e-1 -pente2*1;corr Di=(pente2*ratio2) + b2;


pente1=(10-7)/(1 - 5);b1=7-pente1*5;corr De=(pente1*ratio2) + b1;%pente2=(7e-1 - 1.1)/(1 - 5);b2=1.1-pente2*5;corr Di=(pente2*ratio2) + b2;


pente1=(7-3)/(5-10);b1=3-pente1*10;corr De=(pente1*ratio2) + b1;%pente2=(1.1-1.2)/(5-10);b2=1.2-pente2*10;corr Di=(pente2*ratio2) + b2;


pente1=(3-2.5)/(10 - 50);b1=2.5-pente1*50;corr De=(pente1*ratio2) + b1;%pente2=(1.2-1.3)/(10 - 50);b2=1.3-pente2*50;corr Di=(pente2*ratio2) + b2;


pente1=(2.5-2.1)/(50 - 100);b1=2.1-pente1*100;corr De=(pente1*ratio2) + b1;

195


%pente2=(1.3-1.3)/(50 - 100);b2=1.3-pente2*100;corr Di=(pente2*ratio2) + b2;



endif ( ratio2>500 & ratio2<=1e3)

pente1=(1.9-1.5)/(500 - 1e3);b1=1.5-pente1*1e3;corr De=(pente1*ratio2) + b1;%pente2=(1.15-1.1)/(500 - 1e3);b2=1.1-pente2*1e3;corr Di=(pente2*ratio2) + b2;


pente1=(1.5-1.1)/(1e3 - 5e3);b1=1.5-pente1*1e3;corr De=(pente1*ratio2) + b1;%pente2=(1.1-1)/(1e3 - 5e3);b2=1.-pente2*5e3;corr Di=(pente2*ratio2) + b2;


pente1=(1.1-1)/(5e3 - 1e4);b1=1.-pente1*1e4;corr De=(pente1*ratio2) + b1;%pente2=(1.-1.)/(5e3 - 1e4);b2=1.-pente2*1e4;corr Di=(pente2*ratio2) + b2;

endif ratio2>1e4

corr De=1;corr Di=1;

end

196

Appendix D

Publications related to this thesis

D.1 Publications related to Chapter 4

• Mikikian, M.; Cavarroc, M.; Couedel, L. & Boufendi, L., “Low frequency instabilities

during dust particle growth in a radio-frequency plasma”, Phys. Plasmas, 13, 092103

(2006)

• Mikikian, M.M.; Couedel, L.L.; Cavarroc, M.M.; Tessier, Y.Y. & Boufendi, L.L.,“Plasma

Emission Modifications and Instabilities Induced by the Presence of Growing Dust Parti-

cles”, IEEE Trans. Plasma Sci., In press (2008)

197

Low frequency instabilities during dust particle growthin a radio-frequency plasma

Maxime Mikikian,a! Marjorie Cavarroc, Lénaïc Couëdel, and Laïfa BoufendiGREMI, Groupe de Recherches sur l’Energétique des Milieux Ionisés UMR6606,CNRS/Université d’Orléans, 14 rue d’Issoudun, BP6744, 45067 Orléans Cedex 2, France

!Received 1 June 2006; accepted 26 July 2006; published online 6 September 2006"

In this paper, instabilities appearing in a dusty plasma are experimentally investigated. These low

frequency self-excited instabilities appear during dust particle growth and are characterized by a

frequency spectrum evolving during this process. The onset, the time evolution and the main

characteristics of these instabilities are investigated thanks to electrical and optical measurements.

Both signals show a clear evolution scheme with a well-defined succession of phases. From the

beginning to the end of this scheme, regular oscillations and/or chaotic regimes are observed.

Finally, instabilities stop when the dust particle size reaches a few hundreds of nanometers and a

stable three-dimensional dust cloud is obtained. A dust-free region called void is then usually

observed in the plasma center. © 2006 American Institute of Physics. #DOI: 10.1063/1.2337793$

I. INTRODUCTION

In a plasma, dust particles can be grown using reactive

gases like silane1–3

or by ion bombardment on materials.4–9

This formation is actively studied due to strong conse-

quences that can arise from the presence of dust in plasma

processing reactors10,11

for microelectronics where cleanli-

ness is a major requirement.12

Furthermore, the interest is

actually increasing due to the fact that nanometer dust par-

ticles can be useful for industrial applications like solar

cells13

or memories.14

Laboratory dusty plasmas are also a

very efficient way to produce and study astrophysical dusty

media like planet atmospheres.15

In capacitively coupled

radio-frequency discharges, a dense cloud of submicrometer

dust particles filling the whole space between the electrodes

can be obtained. During their growth, dust particles acquire a

negative charge by attaching more and more electrons.16–18

Consequently, the growth can be studied through its influ-

ence on the plasma characteristics and in particular on the

discharge current harmonics. Indeed, the current third har-

monic is a robust tool to accurately follow and identify

growth steps.19

Dust particle growth strongly affects plasma

properties and can induce plasma instabilities. Self-excited

instabilities have been observed in silane plasmas20

!frequen-

cies of few kHz for dust particles of few nanometers" and

also in dusty plasmas produced by sputtering a carbon

target7,8

!frequencies around 100 Hz for dust particles around

hundred nanometers". In this last experiment, the authors de-

scribe two different instability modes called the filamentary

mode and the great void mode. The first one appears ap-

proximately 1 min after the plasma ignition and is character-

ized by a broadband spectrum centered around 100 Hz. It

corresponds to a beamlike striation of dust density and

plasma glow. As dust particles are still growing, this stage is

then followed by the great void mode, which corresponds to

the formation of a dust-free region called void7,9,21–23

and its

rotation in a horizontal plane in between the electrodes. The

void region is due to the equilibrium between various forces

acting on the negatively charged dust particles. These forces

are principally due to ion drag, electric fields, thermal gradi-

ents, gas flow, and gravity, and they define the cloud shape.

The ion drag force is presently actively studied because it is

suspected to be the main responsible of the void

formation,8,24–33

pushing the dust particles away from the

discharge center. In this paper, observations and analyses of

instabilities appearing during dust particle growth are per-

formed. These instabilities look like the filamentary and

great void modes in some stage of their evolution. Neverthe-

less, their time evolution is much more complicated and

seven different regimes can be identified. Indeed, well-

defined frequencies are observed and the associated spectrum

strongly evolves during the growth process. The instability

complex shape underlines the coexistence of different phe-

nomena that could interact and give regular or chaotic oscil-

lations. The instability time evolution reveals drastic and

sudden changes in shape and frequency that could not be

easily correlated to dust particle size and density. A detailed

analysis of the instability main characteristics is performed

in order to underline and bring to the fore the complex phe-

nomena that could arise in a plasma containing changing

dust particles.

II. EXPERIMENTAL BASIS

A. Experimental setup

The work presented here is performed in the PKE-

Nefedov !Plasma Kristall Experiment" chamber designed for

microgravity experiments.34

The experimental setup consists

of a parallel plate rf discharge where an argon plasma

!0.2–2 mbar" is created in a push-pull excitation mode

!0–4 W". The electrodes are separated by 3 cm and their

diameter is 4 cm. The dust cloud is illuminated by a thin

laser sheet perpendicular to the electrodes and the scattered

light is recorded at 90° with two standard charge-coupled

device !CCD" cameras at 25 images per second. In the firsta"Electronic mail: [email protected]

PHYSICS OF PLASMAS 13, 092103 !2006"

1070-664X/2006/13"9!/092103/8/$23.00 © 2006 American Institute of Physics13, 092103-1

Downloaded 06 Sep 2006 to 194.167.30.129. Redistribution subject to AIP license or copyright, see http://pop.aip.org/pop/copyright.jsp

stages of growth, dust particles are very small and the scat-

tered light is maximum in the laser direction. Consequently,

a third camera is watching the cloud with an angle lying

approximately between 20° and 30° with respect to the inci-

dent laser direction. Instabilities are characterized by two

different diagnostics. First, the time evolution of the ampli-

tude of the discharge current fundamental harmonic is re-

corded. This electrical measurement is representative of glo-

bal changes in plasma properties, especially concerning the

electron density. The second diagnostic is based on spatially

resolved optical measurements. Five optical fibers are hori-

zontally aligned !5 mm in between each fiber" and record the

total plasma light with a spatial resolution of about 3 mm.

This diagnostic gives local measurements that are of interest

to detect any plasma motion or localized changes in order to

better understand the integrated electrical measurements.

B. Dust particle growth

Dust particles are grown by sputtering a polymer layer

deposited on the electrodes and coming from previously in-

jected dust particles !3.4 !m, melamine formaldehyde". A

typical working pressure is around 1.6 mbar and rf power is

about 2.8 W. With these parameters dust particles start to be

detected by the cameras approximately 1 min after the

plasma ignition. A more complete description of the experi-

mental setup and some results concerning dust particle

growth are given in Ref. 9. The growth process leads to

various shapes of the dust cloud !few layers, domelike shape,

three-dimensional dense cloud" due to the fact that we do not

control precisely the size and density of grown dust particles.

Indeed, the growth process seems to be highly sensitive to

gas purity. This effect is amplified by the fact that experi-

ments are performed at static pressure !no gas flow". At least

1 h of pumping between each experiment is needed to elimi-

nate species formed during the previous run and/or coming

from the outgassing of the walls and/or from the sputtered

matter. This behavior is well observed on successive runs by

using emission spectroscopy. Impurities like N2 and OH are

observed and their effect on dust particle formation is under

investigation. Until now it is not clear if their presence pre-

vents the sputtering or if the sputtering occurs but the growth

cannot takes place. This effect of the gas purity has been

previously reported in similar experiments.8

Possible precur-

sors like carbon molecules C2, CN, and CH appear and the

C2 molecule seems to be a good indicator of dust

formation.35

A very high dust density is obtained when base pressure

before an experiment is sufficiently low !few 10−6 mbar". A

base pressure of few 10−5 mbar drastically reduces the

amount of grown dust particles. This base pressure depen-

dence is even more drastic on the instability appearance. In-

deed, in our experimental conditions, they are observed only

if the base pressure is lower than 4"10−6 mbar. Conse-

quently, unstable conditions depend on gas purity and/or on

dust particle density.

III. DUST PARTICLE GROWTH INSTABILITIES

Dust particle growth instabilities !DPGI" typically ap-

pear a few tens of seconds after plasma ignition. Their ap-

pearance is well observed on both electrical and optical

measurements.36,37

Due to their low frequency and strong

amplitude, the beginning of DPGI is also well detected by

the naked eye looking at the plasma glow.

When dust particles are growing in a plasma, the ampli-

tude of the discharge current fundamental harmonic is de-

creasing due to the electron attachment on the dust particle

surface. This decrease is well observed in Fig. 1 from around

20 s !plasma ignition" to 800 s. The appearance of DPGI is

also clearly evidenced by an increase of the fluctuation am-

plitude becoming a real oscillation. DPGI can last several

minutes !more than 12 min in Fig. 1" and different regimes

are evidenced. Their starting time is slightly changing from

one experiment to another due to differences in dust particle

density, which is strongly dependent on gas purity. Neverthe-

less, statistics performed on several experiments allow to de-

duce some general behavior. For example, the pressure de-

pendence is shown in Fig. 2. A decreasing exponential

variation of DPGI appearance time as a function of pressure

is observed: the higher the pressure, the shorter the appear-

ance time. This variation underlines the relation between the

instability appearance and dust particle density. Indeed,

DPGI are observed only when a huge dust particle density is

obtained. From our previous experiments,35

we define the

threshold pressure permitting dust particle growth at around

1.2 mbar. At higher pressures, a shorter delay before forma-

tion of a dense cloud is measured. This delay is well evi-

denced in Fig. 2. At 1.4 mbar, nearly 2 min are necessary to

initiate DPGI while only 40 s are required at 1.8 mbar. Two

possibilities can explain this delay at low pressures: either a

longer time is required to attain the critical dust density nec-

essary to initiate DPGI or the dust density is lower and for

DPGI to begin it needs bigger dust particles !i.e., longer

FIG. 1. !Color online" Time evolution of the amplitude of the discharge

current fundamental harmonic during dust particle growth. Instabilities ap-

pear around 40 s after plasma ignition.

092103-2 Mikikian et al. Phys. Plasmas 13, 092103 "2006!


time". These explanations are consistent with another obser-

vation: the lower the rf power, the longer the appearance

time.

IV. PHASE IDENTIFICATION: ELECTRICALAND OPTICAL MEASUREMENTS

As can be seen in Fig. 1, different regimes characterize

DPGI. These different phases are easily brought to light by

performing the same electrical measurements but in ac mode

in order to improve oscilloscope vertical resolution. The re-

sulting curve is presented in Fig. 3. The beginning of DPGI

is detected around 40 s and clear phases, numbered from 1 to

7, are observed. These different phases are better evidenced

by performing Fourier analysis of the electrical signals. A

typical spectrogram is given in Fig. 4. In order to emphasize

small ordered domains, the spectrogram intensity has been

normalized inside each 100 s range !from 0 to 100 s inten-

sity has been normalized to its maximum value inside this

time domain and so on". The same Fourier analysis per-

formed on spatially resolved optical measurements shows

roughly the same spectrogram features and phases. From

Figs. 3 and 4 we can identify seven different regimes !a

precise analysis of these different phases will be presented in

the following sections":

• Three ordered phases P1, P2, P3 !from %40 s to

%80 s".

• Chaotic phase P4 !from %80 s to %405 s".

• High frequency phase P5 !from %405 s to %435 s".

• Chaotic phase becoming more and more regular P6

!from %435 s to %600 s".

• Regular oscillation phase P7 !from %600 s to %680 s".

All these phases are observed on both electrical and optical

measurements, confirming their real correlation to an un-

stable state of the plasma-dust particle system. The global

scheme of these instabilities is different from the one de-

scribe in Refs. 7 and 8, where a filamentary mode !that could

be related to our chaotic regime" is followed by a regular

phase !great void mode". In our experiment, instabilities be-

gin with regular oscillations !P1, P2, P3" followed by a long

chaotic regime !P4". This chaotic phase !P4" suddenly ends

with a high-frequency phase !P5" and starts again !P6", be-

coming more and more ordered. Finally, the system reaches a

regular phase !P7" that can be sustained for a long time !in

the example given here the plasma has been switched off at

680 s".

A. First three ordered phases

DPGI begin with a succession of three ordered phases

separated by clear transitions !Fig. 5". The P1 and P2 phases

are short and are not detected in all experiments while the P3

phase lasts longer and is regularly observed. The three

phases are well separated and evolve as a function of time.

To explain the observed transitions between phases, it is nec-

essary to also analyze the corresponding time series. As an

example, Fig. 6 represents an electrical measurement ob-

tained with a very short P2 phase in order to illustrate on the

FIG. 2. !Color online" Instability appearance time as a function of argon

pressure.

FIG. 3. !Color online" Time evolution of the amplitude of the discharge

current fundamental harmonic during dust particle growth !ac component".

Instabilities appear around 40 s after plasma ignition. Successive phases are

numbered from 1 to 7.

FIG. 4. !Color online" Spectrogram of electrical measurements correspond-

ing to Fig. 3 and describing the frequency evolution of DPGI as a function

of time.

092103-3 Low frequency instabilities during dust particle growth¼ Phys. Plasmas 13, 092103 "2006!


same figure the P1–P2 and P2–P3 transitions. In Fig. 6, the

P1 phase is characterized by wide separated peaks with a

mean frequency of about 40 Hz. The transition from P1 to P2

corresponds to the growth of two small peaks between these

higher amplitude patterns. The Fourier analysis traduces this

change by a frequency decrease from about 40 Hz in P1

to about 30 Hz in P2. The small peaks continue to grow

!P2 phase" and the higher amplitude ones decrease. Finally,

all peaks reach the same amplitude characterizing the P3

phase. The frequency of the P3 phase is then approximately

three times the P2 frequency !around 94 Hz". The P3 phase

is robust !nearly always observed in our experiments" and

lasts a sufficiently long time to evolve with dust particle

growth !Fig. 5". Its frequency time evolution is always the

same, it decreases, reaches a minimum value, and then

slightly increases until DPGI enter in the chaotic regime.

Furthermore, the frequency range of the P3 phase seems to

be linearly dependent on the phase duration: the higher the

frequency, the shorter the time duration. The time evolution

of these three ordered phases is better observed on the fre-

quency harmonics appearing due to the nearly sawtooth

shape of the electrical signals !Fig. 6".

Optical measurements recording the plasma light !inte-

grated on all wavelengths" at different positions also show

three ordered phases !Fig. 7". Nevertheless, these P1, P2, and

P3 phases have different characteristics than the ones ob-

served on electrical measurements. Concerning the oscilla-

tion shape, the transition between the P1 and P2 phases is

here revealed by an increase of oscillation amplitude !Fig. 7".

The P3 phase is strongly marked by a clear amplitude modu-

lation, which is better evidenced near the plasma edge than

in the plasma center. Concerning the oscillation frequency,

some small discrepancies with electrical measurements are

also observed, especially on the P3 phase !Figs. 5 and 8".

Indeed, the P1 and P2 phases are similar but when DPGI

enter in the P3 phase, the typical frequency is not tripled, like

in electrical measurements, but remains around the same

value than during the P2 phase. Furthermore, optical mea-

surements of this P3 phase are slightly different depending

on the observed plasma region. The signal recorded by the

optical fiber watching the plasma center #Figs. 8!a" and 8!c"$

is not exactly the same as the one recorded near the plasma

edge #Figs. 8!b" and 8!d"$. Indeed, the central fiber gives a

main frequency around 26 Hz and a small amplitude compo-

nent around 31 Hz #Fig. 8!c"$. The near plasma edge fiber

gives the same frequencies but also additional ones: one in

between 26 and 31 Hz and another one around 3 Hz #Fig.

8!d"$ that corresponds to the strong modulation observed in

Fig. 7. These observations underline that some spatial con-

siderations must be made to interpret this P3 phase. The first

local indication concerns the frequency that is three times

lower than for electrical measurements. This point seems to

signify that a phenomenon oscillating at around 94 Hz af-

fects the plasma, and consequently the electrical measure-

ment, but is detected only one time out of three by the optical

fibers. One possible explanation is that a plasma modification

either appears successively in different places, or moves and

comes back in front of an optical fiber with a frequency of

about 30 Hz. This plasma modification appears successively

FIG. 5. !Color online" First three ordered phases of DPGI and transition to

a chaotic regime.

FIG. 6. !Color online" Transitions between the first three ordered phases

observed on electrical measurements.

FIG. 7. !Color online" Transitions between the first three ordered phases

observed on optical measurements performed near the plasma edge.



in distinct locations of the plasma because it is not detected

at 94 Hz by any optical fiber looking at the plasma center or

edge. Investigations concerning this hypothesis are currently

underway using a high speed camera. The second local indi-

cation giving credit to this hypothesis is the difference be-

tween observations given by each optical fiber. Indeed, even

if their Fourier spectrum is nearly similar and shows only

small discrepancies, the recorded time series bring more in-

formation. Optical measurements performed by the near

plasma edge fiber is in phase opposition with the ones in the

center. This behavior, coupled with the fact that a stronger

modulation is observed near the plasma edge, could confirm

a possible motion, or appearance in different places, of a

modified plasma region.

B. Chaotic regime

After three different ordered phases, DPGI enter in a

chaotic regime P4 !Fig. 5 after 80 s" with a strong increase in

DPGI amplitude !Fig. 3". Then, the amplitude slowly de-

creases during the whole phase. P4 is also characterized by

structured oscillations appearing in a transient manner. These

structured regions are identified by some bright spots on

electrical and optical spectrograms. On time series they ap-

pear as bursts of order. In Fig. 9!a", electrical signals reveal a

clear transition between the P3 phase and the chaotic regime.

This regime change is indicated by an arrow and a burst of

order is encircled. A zoom of this burst of order is shown in

Fig. 9!b". During the chaotic regime, structured oscillations

in electrical signals always appear following a three peak

structure #see, for example, between 94 s and 94.04 s in Fig.

9!b"$ that could be related to the three peaks observed in the

P2 or P3 phases. Indeed, it could be a reemergence of these

phases during the chaotic regime. To compare our observa-

tions with the filamentary mode,7,8

a Fourier spectrum of the

whole chaotic regime !from 80 to 405 s" has been performed

in Fig. 10. For electrical signals #Fig. 10!a"$, a noisy main

frequency around 28 Hz is observed and is equal to the fre-

quency at the end of the P2 phase, confirming the possible

reemergence of this phase. This frequency corresponds to the

occurrence frequency of the three peak structure regularly

appearing during the chaotic regime #Fig. 9!b"$. Due to these

structures, multiple frequencies !54 Hz, 83 Hz, and 107 Hz

corresponding to the frequency between two successive

peaks of one single three peak structure" are also detected in

Fig. 10!a". For optical measurements #Fig. 10!b"$, differ-

ences with electrical ones are the same than in the P3 phase:

one optical oscillation corresponds to a three peak structure

and consequently only the 28 Hz frequency appears #Fig.

10!b"$. These spectra are very similar to the one obtained for

the filamentary mode8

even if in our experiments the pres-

ence of more ordered domains is observed. Thus, we can

surmise that the chaotic regime we observe could be similar

to this filamentary mode.

C. High-frequency phase

In some experiments, the chaotic regime is suddenly in-

terrupted by a strong frequency change !at 405 s in Figs. 3

and 4". Indeed, this phase P5 is not always observed but

when it is present, its general characteristics are nearly al-

ways the same. This new phase appears after a continuous

FIG. 8. !Color online" Spectrogram of optical measure-

ments recorded !a" in the plasma center, !b" near the

plasma edge. !c" Zoom of !a", !d" zoom of !b".

FIG. 9. !Color online" Electrical measurements: !a" transition between P3

and P4 phases and encircled burst of order, !b" zoom of the encircled part of

!a".



decrease of the P4 phase amplitude and is defined by a radi-

cal change in DPGI amplitude and frequency. Indeed, DPGI

turn into a low-amplitude and high-frequency !around

500 Hz" oscillations as shown in Fig. 11. This change hap-

pens suddenly when no fast modifications in dust particle

size and density are expected. Furthermore, the phase fre-

quency increases with time. In Fig. 4, this increase is nearly

linear but nonlinear behaviors have also been observed in

certain cases. As previously mentioned, this phase is not al-

ways observed, which means relatively precise conditions

must be fulfilled for its existence. This effect is well con-

firmed in the experiment described in Fig. 12, where this

phase is interrupted before starting again. Small modifica-

tions in plasma or dust particle properties can easily turn the

system from the high-frequency phase to the chaotic one.

Furthermore, this P5 phase can also be transformed in regu-

lar oscillations as shown in Fig. 12 at t=294 s. From these

experimental results it can be assumed that this high-

frequency phase is a particular case of DPGI, obtained in a

tight set of parameters and that can be easily turned in or-

dered or chaotic regimes.

D. Second chaotic regime

After the high-frequency phase, a second chaotic regime

is usually observed. Two different types of this P6 phase are

observed. The most usual is presented in Fig. 4 between 435

and 600 s. It corresponds to a phase similar to P4 but with

more and more ordered regions. Figure 4 clearly shows that

DPGI tend to stabilize to regular oscillations !%600 s". This

transition appears as a small and continuous increase in

DPGI frequency. Nevertheless, some experiments show a

slightly different behavior as in Fig. 13. Instead of slowly

increasing, the frequency decreases and enters !still decreas-

ing" in a regular oscillation phase. This behavior corresponds

to void rotation in a horizontal plane, which is clearly evi-

FIG. 10. !Color online". Fourier spectrum of the first chaotic regime on !a"

electrical measurements, !b" central optical fiber measurements

FIG. 11. !Color online" !a" Appearance of the high-frequency phase in be-

tween two chaotic regimes. !b" Zoom of !a".

FIG. 12. !Color online" Spectrogram of a particular case of the high-

frequency phase. This phase is interrupted by a chaotic regime and is trans-

formed in regular oscillations just before turning again in a chaotic regime.

FIG. 13. !Color online" Spectrogram of a particular case of the second

chaotic regime corresponding to void rotation in a horizontal plane.



denced thanks to CCD camera images. The corresponding

spectrogram is nearly similar to the one obtained in Ref. 8

and related to the great void mode.

E. Final regular oscillation phase

The final step of DPGI often corresponds to a regular

oscillation phase characterized by a spectrogram with a typi-

cal frequency around 18 Hz !Fig. 4" and its harmonics. The

corresponding electrical signal is shown in Fig. 14. The sig-

nal shape is in particular marked by a sharp peak appearing

at each period. This shape is similar to what is obtained

during the “heartbeat” instability,23,38

which corresponds to

regular contractions and expansions of the void size. At this

stage of DPGI, this regular oscillation phase can last for

minutes and is robust, meaning that once it is set up it does

not turn in another regime. Optical measurements performed

in the plasma center show an evolution similar to the elec-

trical one except near the sharp peak region as observed in

Ref. 36. Furthermore, the near edge optical fiber signal is in

phase opposition with the central one !as in Ref. 36", empha-

sizing the similarity with the heartbeat instability.

V. DISCUSSION AND CONCLUSIONS

In this paper, self-excited instabilities induced by the for-

mation of submicrometer dust particles are investigated.

These instabilities appear when a sufficiently high dust par-

ticle density is reached. Instability frequency and shape

change during the growth process and direct correlation with

dust particle size and density cannot be easily deduced. In-

deed, effects of both parameters on discharge behavior are

difficult to separate: a high density of small particles can

have roughly the same effect than a lower density of bigger

particles, assuming that the total collection surface is identi-

cal and consequently the same amount of electrons can be

captured. The only way to conclude on this point is to mea-

sure dust particle size and density but most commonly used

in situ diagnostics cannot be implemented on our experimen-

tal setup. Furthermore, an easy systematic collection of dust

particles coming from one and only one experiment is not

possible in our reactor. DPGI reproducibility can be obtained

only with very low base pressures not reached in the case of

regular reactor opening. By comparing Fig. 1 with electrical

characteristics obtained in silane based discharges,39

we can

speculate that DPGI occur during the “coalescence phase”

and last during the so-called “surface deposition phase.”

These phases40

are characterized by a significant amount of

negatively charged dust particles strongly impacting on

plasma stability. This comparison with dust particle growth

process in silane based chemistry must be taken carefully

because the analogy has not been strictly proven yet. From

visual observations we know that when DPGI occur, dust

particles are not detected yet by our video system and are

certainly smaller than 100 nm. When instabilities stop, the

dust particle size could be of a few hundreds of nanometers.

The instability development follows a well-defined succes-

sion of phases that can be ordered or chaotic. At least seven

different regimes have been evidenced and their occurrence

is strongly dependent on plasma and dust particle properties.

Complex transitions between different ordered phases or be-

tween ordered and chaotic ones have been evidenced and

underline the nonlinear behavior of these instabilities. Two

observed phases seem to correspond to the ones reported in

previous papers.7,8

These phases, called the filamentary mode

and the great void mode, can be a part of a more general

behavior described in this paper. The new phases described

here underline the complexity of the phenomena behind ob-

servations and their high sensitivity to plasma and dust par-

ticle properties. Indeed, the complete scheme with seven

phases can be observed in very specific experimental condi-

tions usually encountered in this work. Striations !in our

case, enhanced ionization regions appearing as bright spots

or filaments" seem to be responsible for the different ob-

served phases. The various regimes could be related to the

way striations appear. Indeed, we can surmise that the or-

dered regimes could be striations appearing periodically and

nearly at the same places. Following this hypothesis, the of-

ten observed three peak structure can correspond to three

different striations. By analogy, chaotic regimes can be due

to randomly appearing !in time and in space" striations. Op-

tical measurements performed in different plasma positions

bring to the fore that these instabilities need to be analyzed

with spatial considerations. Differences with works reported

in Refs. 7 and 8 can be the consequence of different plasma

and/or dust particle properties, geometrical considerations, or

gas flow. Furthermore, our observations show that void for-

mation is not a consequence of these instabilities as proposed

in Ref. 24. The void can be formed with or without instabili-

ties preceding its appearance. In some conditions, the void is

formed before instability begins. In this case, the instability

evolution scheme does not describe all the phases presented

here. Similarities between the final regular oscillation phase

and the heartbeat instability concerning the void region have

been observed. It could indicate that dust particle growth

instabilities and the heartbeat instability could be different

aspects of the same physical phenomena.

FIG. 14. !Color online" Electrical measurements corresponding to the final

regular oscillation phase.



ACKNOWLEDGMENTS

The PKE-Nefedov chamber has been made available

by the Max-Planck-Institute for Extraterrestrial Physics,

Germany, under the funding of DLR/BMBF under Grant No.

50WM9852. The authors would like to thank S. Dozias for

electronic support, J. Mathias for optical support, and Y.

Tessier for experimental support.

This work was supported by CNES under contract No.

793/2000/CNES/8344.

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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE TRANSACTIONS ON PLASMA SCIENCE 1

Plasma Emission Modifications and InstabilitiesInduced by the Presence of Growing Dust Particles

Maxime Mikikian, Lénaïc Couëdel, Marjorie Cavarroc, Yves Tessier, and Laïfa Boufendi

Abstract—Formation of dust particles in a plasma can stronglychange its properties due to electron attachment on dust surface.An easy way to detect dust formation is to analyze modifications ofthe plasma emission. In this paper, changes in the plasma emissionare related to the growth of dust particles. We particularly showthat dust formation induces low-frequency plasma instabilities.Another interesting induced effect is the formation of an enhancedemission region which is dust free and usually named “void.”

Index Terms—Complex plasma, dust particle growth, dustyplasma, instabilities, oscillations, plasma emission, void.

PLASMAS ARE widely encountered in nature and indus-

tries. In many environments, these plasmas also contain

dust particles (i.e., solid bodies from a few nanometers to

centimeters) and are thus called dusty (or complex) plasmas. In

astrophysics, these media are found in comet tails or planetary

atmospheres. In artificial plasmas, these dust particles are fatal

for processes in microelectronics (where cleanliness is needed)

and for fusion devices like ITER (International Thermonuclear

Experimental Reactor). Nevertheless, in nanotechnology, these

media are useful in building nanostructured films or nanometer-

size devices. Dust particles can be produced from reactive gases

(silane and methane) or material sputtering injecting molecular

precursors or solid bodies in the plasma. Molecular precursors

coming from reactive gases or sputtering thus initiate a complex

succession of chemical and physical reactions, leading to the

growth of dust particles. As these dust particles are growing,

they attach plasma electrons, leading to a disturbance of plasma

equilibrium when the dust particle density is high [1]–[3].

In this paper, modifications of plasma emission and self-

excited instabilities induced by the growing particles are

visually analyzed. These experiments are performed in the

PKE-Nefedov reactor [4] where dust particles are grown by

sputtering a polymer layer exposed to a low-pressure radio-

frequency (13.56 MHz) argon plasma. The capacitively coupled

discharge is maintained by parallel electrodes of 4 cm in diam-

Manuscript received November 28, 2007; revised February 4, 2008. Thiswork was supported by the CNES under Contract 02/CNES/4800000059.

M. Mikikian, M. Cavarroc, Y. Tessier, and L. Boufendi are with theGroupe de Recherches sur l’Energétique des Milieux Ionisés, UMR6606,CNRS/Université d’Orléans, 45067 Orléans Cedex 2, France (e-mail: [email protected]).

L. Couëdel is with the Groupe de Recherches sur l’Energétique des MilieuxIonisés, UMR6606, CNRS/Université d’Orléans, 45067 Orléans Cedex 2,France, and also with the School of Physics, A28, The University of Sydney,Sydney, N.S.W. 2006, Australia.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2008.920946

Fig. 1. Plasma emission evolution induced by the presence of dust particles:(a) 5 s after ignition (AI): No dust particles; (b) 1 min AI: Beginning ofinstabilities (striationlike) induced by the growing dust particles; (c) 2 minAI: Instabilities with increasing intensity; (d) 7 min AI: Plasma emissionenhancement due to the presence of bigger dust particles; instabilities consistof a single enhanced emission region (void) rotating around the vertical axis ofsymmetry; (e) 15 min AI: Plasma emission enhancement and less off-centeredrotation; and (f) 16 min AI: Centered void region with oscillating size.

eter separated by 3 cm. The typical pressure is about 1.4 mbar,

and the injected power is 3.5 W. Plasma emission is recorded

by using a standard color charge-coupled device camera at

25 frames/s. Simultaneously, the discharge current is monitored

to identify the growing step and the occurring instability type.

Fig. 1 shows the plasma emission at different instants after

plasma ignition. Fig. 1(a) has been taken 5 s after plasma

ignition and is considered to be representative of a dust-free

plasma as the growing kinetics is slow in our experimental

conditions. This figure is thus our reference for the following

discussion. It clearly shows the two presheath regions of en-

hanced emission and the plasma column. Fig. 1(b) corresponds

to the beginning of self-excited instabilities induced by the

high density of negatively charged dust particles [5] disturbing

the plasma electrical equilibrium. These dust particle growth

0093-3813/$25.00 © 2008 IEEE


2 IEEE TRANSACTIONS ON PLASMA SCIENCE

instabilities [1] are composed of a complex succession of well-

defined phases. In Fig. 1(b), the instability is characterized by a

striationlike behavior [1], [6], and two slightly brighter regions

appear on both sides of the center. In this regime, bright regions

appear alternatively in different places of the discharge. On

electrical measurements, the instability signature corresponds

to regular oscillations of a few tens of hertz. Even if the sam-

pling rate of the camera is not sufficient to finely analyze this

oscillation, it gives important information on plasma evolution

as a major complement of electrical measurements. After this

relatively regular regime, a chaotic phase occurs with bright

regions appearing in a more erratic manner. This chaotic regime

is partly characterized by a slight increase in the intensity of the

bright regions, as shown in Fig. 1(c). This regime can last a

few minutes and slowly tends toward a more and more ordered

phase described by a single bright region that can be, at this

stage, easily linked to a dust-free region called “void.” This

region, which can appear with or without precursory instability,

is related to the competition of an inward electrostatic force

and an outward ion drag force acting on dust particles. Once

a clear void region exists in a dusty plasma, different kinds of

instabilities are observed. In Fig. 1(d), taken around 7 min after

plasma ignition, this void rotates around the vertical symmetry

axis of the discharge. The void is clearly identifiable slightly

on the right-hand side. The dust cloud at this instant is made

of a high density of big dust particles (a few hundreds of

nanometers). Their influence on the plasma is important and

results in a strong increase of the plasma emission as observed

in the void and above the presheath regions. The rotation

slows down, and the rotation radius decreases [Fig. 1(e)]. Then,

the enhanced emission void region stabilizes its position in

the discharge center. At this time, another type of instability

is able to disturb the plasma and the dust cloud. It consists

of successive contraction and expansion of the void size [7]

[Fig. 1(f)]. Due to its characteristic behavior, this instability has

been named the “heartbeat” instability.

In this paper, we presented an easy visual indicator of the

presence of a high dust particle density in a plasma. Changes

in plasma emission intensity and appearance of self-excited

instabilities easily detected by cameras are consequences of the

drastic loss of free electrons to the dust particle surface.

ACKNOWLEDGMENT

The PKE-Nefedov chamber was made available by the Max-

Planck-Institute for Extraterrestrial Physics, Germany, under

the funding of DLR/BMBF under Grants No. 50WM9852.

REFERENCES

[1] M. Mikikian, M. Cavarroc, L. Couëdel, and L. Boufendi, “Low frequencyinstabilities during dust particle growth in a radio-frequency plasma,” Phys.Plasmas, vol. 13, no. 9, p. 092 103, Sep. 2006.

[2] M. Cavarroc, M. C. Jouanny, K. Radouane, M. Mikikian, and L. Boufendi,“Self-excited instability occurring during the nanoparticle formation in anAr-SiH4 low pressure radio frequency plasma,” J. Appl. Phys., vol. 99,no. 6, p. 064 301, 2006.

[3] M. Cavarroc, M. Mikikian, G. Perrier, and L. Boufendi, “Single-crystalsilicon nanoparticles: An instability to check their synthesis,” Appl. Phys.

Lett., vol. 89, no. 1, p. 013 107, 2006.[4] A. P. Nefedov et al., “PKE-Nefedov: Plasma crystal experiments on the

international space station,” New J. Phys., vol. 5, no. 3, pp. 33.1–33.10,2003.

[5] L. Couëdel, M. Mikikian, L. Boufendi, and A. A. Samarian, “Residual dustcharges in discharge afterglow,” Phys. Rev. E, Stat. Phys. Plasmas Fluids

Relat. Interdiscip. Top., vol. 74, no. 2, p. 026 403, 2006.[6] G. Praburam and J. Goree, “Experimental observation of very low-

frequency macroscopic modes in a dusty plasma,” Phys. Plasmas, vol. 3,no. 4, pp. 1212–1219, Apr. 1996.

[7] M. Mikikian, L. Couëdel, M. Cavarroc, Y. Tessier, and L. Boufendi, “Self-excited void instability in dusty plasmas: Plasma and dust cloud dynamicsduring the heartbeat instability,” New J. Phys., vol. 9, no. 8, p. 268, 2007.

D.2 APPENDIX D. PUBLICATIONS


• Mikikian, M.; Couedel, L.; Cavarroc, M.; Tessier, Y. & Boufendi, L., “Self-excited void

instability in dusty plasmas: plasma and dust cloud dynamics during the heartbeat insta-

bility”, New J. Phys., 9, 268 (2007)

• Mikikian, M.; Cavarroc, M.; Couedel, L.; Tessier, Y. & Boufendi, L. “Mixed-Mode Oscil-

lations in Complex-Plasma Instabilities” Physical Review Letters, 100, 225005 (2008)

203

T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s

New Journal of Physics

Self-excited void instability in dusty plasmas:

plasma and dust cloud dynamics during the

heartbeat instability

M Mikikian, L Couëdel, M Cavarroc, Y Tessier and L Boufendi

GREMI, Groupe de Recherches sur l’Energétique des Milieux Ionisés,

UMR6606, CNRS/Université d’Orléans, 14 rue d’Issoudun,

BP6744, 45067 Orléans Cedex 2, France

E-mail: [email protected]

New Journal of Physics 9 (2007) 268

Received 30 March 2007

Published 15 August 2007

Online at http://www.njp.org/

doi:10.1088/1367-2630/9/8/268

Abstract. When a three-dimensional dust cloud is present in a plasma, a dust-

free region, called a void, is usually obtained in the plasma centre. Under certain

conditions, this region exhibits a self-excited unstable behaviour consisting

of successive contractions and expansions of its size. In this paper, this low

frequency instability (few Hz), called a ‘heartbeat’, is characterised by various

diagnostics. Electrical and optical measurements both correlated with high

speed imaging brought to the fore the main features of this instability. Forces

involved in the void existence are an inward electrostatic force and an outward

ion drag one. The force balance ensures an open void but this equilibrium

can be disturbed, leading to the observed instabilities. As these forces are

strongly dependent on local ionisation conditions, correlations between physical

processes in the plasma volume and the dust cloud motion are investigated

through experimental results.

New Journal of Physics 9 (2007) 268 PII: S1367-2630(07)47353-01367-2630/07/010268+18$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2

Contents

1. Introduction 2

2. Experimental set-up 3

2.1. Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2. Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3. Dust cloud and plasma glow behaviours near the void region 5

3.1. Sequences with a low repetition rate . . . . . . . . . . . . . . . . . . . . . . . 5

3.2. Sequences with a high repetition rate . . . . . . . . . . . . . . . . . . . . . . . 8

3.3. Asymmetry in void contraction . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4. Plasma glow behaviour in the whole discharge 11

4.1. Sequences with a low repetition rate . . . . . . . . . . . . . . . . . . . . . . . 11

4.2. Sequences with a high repetition rate . . . . . . . . . . . . . . . . . . . . . . . 13

5. Discussion and conclusion 16

Acknowledgments 17

References 17

1. Introduction

Dusty plasmas are relatively complex systems where new phenomena arise from the presence

of solid dust particles trapped inside the plasma. Usually, this trapping is a consequence of the

negative charge acquired by the dust particles [1]–[6] immersed in a medium partly composed

of free electrons. The full dust particle dynamics is then described by charge dependent

forces (mainly electric and ion drag forces) and other ones like gravity, neutral drag and

thermophoresis. Relative amplitudes of these forces are strongly dependent on the dust particle

size. In the micrometre range, dust particles are usually directly injected in the plasma volume.

Their weight is non-negligible and some phenomena are only observed under microgravity

conditions [7, 8]. Nevertheless, their big size allows an easy and precise visualisation in

laboratory experiments. In the submicrometre range, dust particles are usually grown in the

plasma using methods based on reactive gases or material sputtering.

Reactive gases are used due to their ability to form dust particles by following a rather

complicated succession of growth phases. One of the most studied and well-known reactive gas

is silane (SiH4) [9]–[15]. The interest for silane based chemistry is mainly due to its implication

in microelectronics and/or nanotechnology. Indeed, in the late 1980s, dust particle formation

in the gas phase has been evidenced in reactors used for silicon based device fabrication

[9, 16]. In some processes, cleanliness is a major requirement and a lot of studies began for

understanding dust particle formation and growth in order to avoid their deposition. More

recently, silicon nanocrystal formation [17] became of high interest for their incorporation

in thin films in order to improve their properties. Indeed, their use in solar cell technology

enhances optoelectronic properties of deposited films [18]. Single electron devices (SED) like

transistors [19] or memories [20] can also be built, thanks to these silicon nanocrystals. Dust

particle formation in hydrocarbon based gases like methane (CH4) [21]–[23] or acetylene

(C2H2) [21, 22, 24, 25] has also been studied. Indeed, CH4 and C2H2 are used for deposition

New Journal of Physics 9 (2007) 268 (http://www.njp.org/)

3

of diamond-like carbon (DLC) films [26] or nanocrystalline diamond [27] used in industry for

their unique properties like extreme hardness. Hydrocarbon gases are also of great interest for

the astrophysical community dealing with planet atmospheres like Titan, where dust particles

are created from a mixture of methane and nitrogen [28].

Another process for dust particle production is material sputtering [29]–[32]. This

phenomenon can appear in industrial reactors and also in fusion devices [33] like Tore

Supra [34] where graphite walls can be severely eroded by ion bombardment. Produced dust

particles can strongly limit performances of the fusion plasma. This aspect is of great importance

for the future ITER reactor.

Dust particles levitating in the full plasma volume can be obtained under microgravity

(for micrometre dust particles) or in the laboratory (for submicrometre dust particles).

Usually, the three-dimensional dust cloud exhibits a specific feature consisting of a

centred egg-shaped region free of dust particles. This region, called a ‘void’ [7, 31, 32],

[35]–[47], seems to be controlled by an equilibrium between the charge dependent

forces (inward electrostatic and outward ion drag forces). This stable state is sometimes

disturbed and void size oscillation can occur. This self-excited instability is named the

‘heartbeat’ instability [36, 46, 48, 49]. It has been previously observed during microgravity

experiments with injected micrometre dust particles and has since also been studied in the

laboratory with grown submicrometre dust particles [49]–[52]. In these last experiments,

investigations of the heartbeat instability have been performed using both electrical and

spatially resolved optical diagnostics. These measurements allowed to identify complex

behaviours in recorded signals roughly related to dust particle motion. Furthermore,

the threshold behaviour of this self-excited instability has been brought to the fore

[51, 52]. Nevertheless, the presence of a sharp peak in electrical measurements and its absence

in optical ones was still unclear. Further investigations have been conducted using high speed

imaging to record dust particle cloud motion. These preliminary results gave first insights

concerning dust cloud motion and void size evolution correlated with electrical signals [51,

52]. In the present paper, new results on the heartbeat instability are obtained thanks to a second

high speed imaging experiment. Behaviour of both dust cloud and plasma glow is investigated

and correlations between both types of data are deduced. These results are compared to

electrical measurements and previous spatially resolved optical ones. The void dynamics during

the instability is now better characterised and, for example, the sharp peak presence can be

correlated with plasma glow changes.

2. Experimental set-up

2.1. Reactor

Experiments are performed in the Plasma Kristall Experiment (PKE)-Nefedov chamber [8, 32].

The plasma is produced by a capacitively coupled radio frequency (rf ) discharge operating

in push–pull mode at 13.56MHz. The planar parallel electrodes are 4 cm in diameter and are

separated by 3 cm. In this chamber, argon is introduced to a typical pressure of 1.6mbar and

the plasma is ignited with a rf power of typically 2.8W. Dust particles are grown by sputtering

a polymer layer deposited on the electrodes and constituted of previously injected melamine

formaldehyde dust particles.


4

2.2. Diagnostics

Grown dust particles are observed by laser light scattering using a thin laser sheet produced by

a laser diode at 685 nm and three standard charge coupled device (CCD) cameras (25 frames

per second) equipped with narrow-bandwidth interference filters. Two of these cameras record

the scattered light at 90 with different magnifications. The third one looks at the cloud with an

angle lying approximately between 20 and 30 with respect to the incident laser direction. This

camera allows the observation of grown dust particles when their size is too small to observe

them at 90.

Plasma and dust particle cloud behaviours are strongly linked and a modification in one of

these media is usually reflected in the other. Therefore, diagnostics able to analyse both systems

are used in order to better characterise the heartbeat instability.

The first one consists of the measurement of the time evolution of the amplitude of the

discharge current fundamental harmonic. Indeed, this diagnostics is representative of global

changes in plasma properties and can be related to electron density variations. It allows to

monitor the total current during the various phases of the instability very easily. Thus, it is

used as our reference diagnostics. It shows that the heartbeat instability has different signatures

depending on the pressure, power and dust particle density [49].

The second diagnostics is based on spatially resolved optical measurements performed in

two different ways. The time evolution of either the intensity of an argon line or the total plasma

light is recorded in different plasma regions. For the total plasma light recording, five optical

fibres are horizontally aligned (separated by 5mm) with a spatial resolution of about 3mm. This

diagnostic is complementary to the electrical one because it can detect any plasma motion or

local behaviours that cannot be evidenced by a global measurement.

The third diagnostics consists of high speed imaging. The observed instability oscillates

at relatively low frequencies (10–200Hz) but too high to be finely characterised by standard

CCD cameras at 25 frames per second. Indeed, as previously observed [49], some features

of the instability, like the sharp peak, take place during a typical time scale of about 1ms. A

high speed video camera system with 1789 frames per second (Mikrotron MC 1310) is used

to observe either the plasma glow or the dust cloud. Thus, two successive frames are separated

by around 560µs. For the last observation, the high speed camera takes the place of the third

standard CCD camera in order to record more light scattered by the dust cloud.

As already mentioned, the heartbeat instability has different signatures and measurements

show a wide variety of frequencies and shapes. In the following sections, both dust cloud

and plasma glow will be analysed in two different cases considered as sufficiently different

to cover a wide range of experimental observations: a low repetition rate one (contraction–

expansion sequences are well separated and original conditions, i.e. stable void, are nearly

restored between each sequence) and a high repetition rate one (continuous motion of the

void). The high repetition rate occurs when the instability is well established (higher values

of injected rf power) whereas low repetition rate occurs at lower powers when the instability

tends to slow down before stopping. Plasma glow will be mainly discussed through its intensity

(thus ‘glow intensity’ is referred as ‘glow’ in the following), except when spatial concerns are

well specified. Following results will mainly concern the fully developed instability. Another

interesting phenomenon, consisting of failed contractions [51, 52] appearing near the instability

existence threshold, will be analysed in a further article.


5

30 42 45 48

50 52 55 60

70 80 90 120

150 180 210 265

(a)

Image number

Colu

mn p

rofi

le (

pix

els)

50 100 150 200 250 300 350 400 450 500 550 600

80

100

120

140

160

180

200

220

240

(b)

Figure 1. Heartbeat instability for a low repetition rate observed on the dust

cloud. (a) Some images of the void region during one contraction–expansion

sequence. The yellow ellipse delineates position of the stable open void. (b) Time

evolution of central column profile (vertical line passing through the void centre)

constructed from the entire video in false colours (from dark red to bright yellow)

(see also movie 1, 3.9MB MPEG).

3. Dust cloud and plasma glow behaviours near the void region

In order to characterise the heartbeat instability, a first approach consists of a direct visualisation

of the void with its surrounding dust cloud. Typical unstable voids have a size of about a few

millimetres. Beyond a certain size, voids are usually stable and no instability has been observed

in the explored parameter range. Indeed, in this case, their stability is not disturbed even by

decreasing the pressure or increasing the power (these changes are known to be able to initiate

the heartbeat instability [49]).

3.1. Sequences with a low repetition rate

A video corresponding to a low repetition rate and showing three successive contraction–

expansion sequences is available in figure 1. Some extracted images from the first sequence

are shown in figure 1(a). Image aspect ratio is distorted due to the view angle compressing the

horizontal direction. Image numbers appear on the top right part of each image. The drawn

ellipse represents the position of the stable open void determined from image 30. Small but

detectable motion starts at image 40 (not shown) and clearly appears at image 42 where dust

cloud boundaries are clearly inside the drawn ellipse. Contraction lasts up to image 55 where the

void reaches its minimum size. From this instant, the empty inner part slowly increases in size

while a light grey corona region (i.e. lower dust density region) starts to be detected in image

80 and slowly reduces in size. These two regions meet and the stable open void state is nearly

reached anew in image 265.

In order to better depict this contraction–expansion sequence, the time evolution of a

column profile is shown in figure 1(b). To construct this false colour image, the vertical line

passing through the void centre is extracted from each video image. Time is represented

by image number in the x-axis and column profile is given in the y-axis. Using the video

as a reference, this representation is of interest in that it gives in one single image a clear


6

characterisation of the instability in both time and space. Low dust density regions appear

dark while high dust density ones appear bright. Three contractions and two expansions are

clearly evidenced and give a frequency of about 7Hz. For the first sequence, images 1–40 show

the higher dust density region usually observed in the void boundaries. Then, the contraction

takes place and the void reaches its minimum size approximately 2.5 times smaller than the

stable void size. At this moment, the two previously described phenomena (size increase of

the dark inner part and size decrease of an intermediate grey corona region) are well resolved

in time. The regular size increase of the inner part can be directly correlated to the void size

increase assuming that the term ‘void’ stands for the null dust density region. From figure 1,

the behaviour of the corona region seems to correspond to the motion of dust particles attracted

to the plasma centre during the contraction and going back to their original stable position.

This motion is characterised by a moving boundary between a region of high dust density and

another one of low dust density. It appears that the speed of this moving boundary decreases as

it approaches the original void position. The dust particles which are the closest to the centre,

react last with the smallest speed. Furthermore, the corona region in its maximum extension is

larger than the size of the stable void indicating that the instability affects the dust cloud over a

long distance and not only in the close vicinity of the void. Collective motions can take place on

long distances due to the strong interaction between dust particles and their strong coupling with

plasma changes. Finally, figure 1(b) shows that inner and corona regions meet (around image

280) and the high dust density boundaries are restored just before another contraction occurs.

To complete these observations, the interference filter is removed from the high speed

camera in order to record plasma glow evolution without changing the camera position. The

obtained image series is acquired approximately 2 s after the one presented in figure 1, checking

that the instability does not change neither in frequency nor in shape. In order to synchronise

dust cloud images with plasma glow ones, small residual plasma glow signal passing through

interference filter is used (slightly visible in figure 1(b) in the black inner part of the void but

not allowing analysis). Electrical measurements are used to check that instability characteristics

remain unchanged. Typical images are presented in figure 2(a). In order to bring out changes,

a reference image is subtracted and false colours are used. Plasma glow images shown in

figure 2(a) correspond to the dust cloud ones shown in figure 1(a). For comparison, the same

ellipse as in figure 1(a) is superimposed on the glow images. By correlating both dust cloud

and glow images, it clearly appears that the void contraction seems to correspond to a glow

enhancement in the plasma centre. Intermediate images of the glow variation are shown in

figure 2(b). It appears that the concerned plasma region is bigger than the original void region

even if maximum glow values are inside the void. This aspect can explain why the light grey

corona region observed in figure 1 has a maximum extension bigger than the stable void. In fact,

the first image showing the glow increase is image 10 in figure 2(b), where the bright region is

already bigger than the drawn ellipse. Due to camera speed limitation, some data are missing

between images 9 and 10. Consequently it cannot be concluded if, at the early beginning, the

enhancement is bounded by the void and then propagates, or if it concerns directly a bigger

size region. After reaching its maximum value (around image 13), plasma glow decreases and

reaches a minimum below the mean glow value (darker image 19 in figure 2(a)). Certainly due

to dust particle inertia, the void continues to shrink (figure 1(a)) but when the central glow

(i.e. ionisation and consequently ion drag force) starts to increase slowly (from image 26 in

figure 2(a) corresponding to image 55 in figure 1(a)), the contraction stops and expansion occurs.


7

1 13 16 19

21 23 26 31

41 51 61 91

121 151 181 236

(a) 9 10 11

12 13 14

15 16 17

(b)

Figure 2. Heartbeat instability for a low repetition rate observed on the plasma

glow. (a) Some images of the void region (in false colours from dark blue to red)

during one contraction–expansion sequence (image 1 corresponds to image 30

of figure 1 and so on). The yellow ellipse delineates position of the stable open

void. (b) Intermediate images of (a) during fast plasma glow modifications.

200

210

220

230

240

200

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220

230

240

5 10 15 20 25 30 35 40

200

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Image number

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(pix

els)

Colu

mn p

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(pix

els)

Colu

mn p

rofi

le

(pix

els)

(a)

(b)

(c)

Figure 3. Heartbeat instability for a low repetition rate: electrical measurements

superimposed on column profiles from (a) only dust cloud; (b) dust cloud and

plasma glow, (c) only plasma glow. Parts (a), (b) and (c) in false colours (from

dark blue to red).

Column profiles of these different series can be compared. Figure 3 shows synchronised

profiles obtained from videos showing (a) the dust cloud (laser and camera with interference

filter), (b) dust cloud and glow (laser and camera without interference filter) and (c) plasma glow

(neither laser nor interference filter). Electrical measurements are superimposed with their mean

value (dotted line) corresponding to the stable open void. In figure 3(c) it appears that there is a

good correlation between plasma glow and electrical measurements as the main variations are

related. The fast signal increase corresponds to glow enhancement. It is followed by a decrease

in both sets of data. A small shoulder appearing in electrical measurements (slope change) is


8

69 71 73 75

77 79 82 86

93 98 103 108

113 118 123 128

(a)

Image number

Colu

mn p

rofi

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pix

els)

20 40 60 80 100 120 140

80

100

120

140

160

180

200

220

240

(b)

Figure 4. Heartbeat instability for a high repetition rate observed on the dust

cloud: (a) some images of the void region (in false colours from dark red to

bright yellow) during one contraction–expansion sequence; (b) time evolution

of central column profile constructed from the entire video (see also movie 2,

2.3MB).

related to a more drastic decrease of the central plasma glow. Then, the following minimum

values of both data are close in time. After this point, the next plasma glow increase towards the

mean state is well observed in electrical measurements slowly tending towards the mean value.

As mentioned, global variations are similar but some discrepancies exist due to the fact that

electrical measurements are integrated on all the plasma volume, and contain information on

plasma variation occurring outside the plasma centre. This aspect will be explored in section 4,

where a precise analysis of the entire plasma glow is performed. Concerning the dust cloud,

the void contraction is related to glow and electrical signal increase (figures 3(a) and (b)),

meaning that it corresponds to ionisation increase in the plasma centre. Nevertheless, it can

be observed that contraction continues even when the electrical signal starts decreasing and

is still not stopped when electrical measurements and plasma glow are below their original

value. It confirms the direct comparison between dust cloud and plasma glow performed through

figures 1 and 2.

3.2. Sequences with a high repetition rate

Analysis of a high repetition rate sequence is more fussy due to the fast and constant motion of

the dust cloud without any clear stable position giving a fixed reference. A movie is available

in figure 4 and in the same way as in the low repetition rate case, extracted images and column

profile are given respectively in figure 4(a) and (b). In order to increase contrast, false colours

are also used for extracted images. The constant motion clearly appears: no horizontal bright

regions, corresponding to high dust density void boundaries, are observed in figure 4(b). This

figure gives a frequency instability of about 31Hz. Furthermore, the inner part of the void

region, which is never filled with dust particles, exhibits only small variations. Indeed, all the

motion takes place in the low dust density region between the dense dust cloud and the void

inner part. From the nearly isolated case (figure 1), this region has been identified to consist

of dust particles going back to their original position. In the high repetition rate case, the new


9

contraction occurs while dust particles from the previous contraction are still inside the void

and try to reach their original position. Indeed, a new contraction starts around image 69 of

figure 4(a) (complex motions do not allow precise determination) while dust particles are still

inside the void region. At this time the plasma glow is close to its maximum value in the centre

(determined from a plasma glow video correlated to the present one). This complex behaviour

leads to the relatively constant regions (black inner part and grey intermediate part) observed in

figure 4(b). The dust cloud is always moving even so spatial dust density appears more or less

constant.

3.3. Asymmetry in void contraction

Another interesting phenomenon concerning dust cloud motion during the heartbeat instability,

is the observable delay appearing in the response of horizontal and vertical directions. Indeed,

in some conditions (currently not clearly defined) the dust cloud shrinks horizontally before

shrinking vertically [51, 52]. As an example, a record containing both dust cloud and plasma

glow information has been performed for a sequence where a stable open void suddenly starts a

size oscillation. First and second contraction–expansion sequences are shown in figure 5 where

a movie can also be found. Image 135 clearly shows the stable open void with the corresponding

drawn ellipse. In image 142 plasma glow starts increasing and reaches its maximum value at

144. In this last image, it appears that the plasma glow enhancement affects a region bigger than

the void size and correlation with the corona region well observed in figure 1 can be suggested

again. This hypothesis is correct assuming that the intensity increase in the surrounding dust

cloud is entirely due to plasma glow changes and not due to any dust density modification.

Thanks to the drawn ellipse, it clearly appears that the void shrinks first in the horizontal

direction (starting from image 143) and then in the vertical one (starting around image 147). As

images are already compressed in the horizontal direction, a detectable motion in this direction

clearly indicates a real dust cloud response and cannot be an optical illusion. The plasma glow

reaches its minimum value in image 148 and, as already observed, the void continues to shrink

to its minimum size in image 155. From this point, the void inner part slowly increases in

size and the grey intermediate region, constituted of dust particles going back to their original

position, becomes gradually visible. The next contraction occurs before the original state has

been completely reached again (image 241).

These various steps are well observed on the column profile shown in figure 6(a). For

better understanding, temporal evolution of plasma glow in the central pixel of the void is

superimposed at the bottom of the image. The stable open void is accurately defined with its

constant size and its high dust density boundaries (between images 100 and 142). When plasma

glow increases, the void shrinks. Its minimum size is reached while plasma glow in the centre

has already reached its minimum value and has started to slowly increase again. Then, the glow

stays relatively constant in the void centre while dust particles are expelled to their original

position. It clearly appears that the void did not reach its original size when the new glow

increase occurs. This effect is traduced, firstly by the dark inner part which does not rejoin

the position of the stable void and secondly by the intermediate corona region which is bigger

than the original void size. This last point means that the ‘wave front’ formed by returning

dust particles is still moving towards the original void boundaries. The speed of this wave

front appears to slow down when approaching the original conditions. Furthermore, it is clearly

visible that the instability is in a setting up phase. Indeed, starting from a stable open void,


10

Figure 5. Simultaneous record of dust cloud and plasma glow in the void region

during the setting up of the heartbeat instability. From image 135: stable open

void (delineated by a pink ellipse) followed by a first contraction, and new

contraction at image 241 while original conditions are not restored (see also

movie 3, 2.6MB).

Image number

Co

lum

n p

rofi

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pix

els)

100 150 200 250 300 350 400

40

60

80

100

120

140

160

(a)

Co

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n p

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pix

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Image number

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pix

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135 140 145 150 155 160 165

20

30

40

50

60

70

(b)

(c)

Figure 6. (a) Column profile (in false colours from dark red to bright yellow)

calculated from image series presented in figure 5 with central plasma glow

evolution superimposed (blue curve corresponding to line 95 of figure 6(a)),

(b) zoom of (a), (c) corresponding line profile.

the characteristics of first and second contraction–expansion sequences are slightly different in

terms of minimum void size and re-opening time duration. Concerning the time delay appearing

in the vertical direction (as observed in figure 5) it can also be evidenced by comparing

column and line profiles extracted from the video. These two profiles are shown respectively

in figures 6(b) and (c). Previous results also showed that some correlation can be found

between the different times of collapse and different slopes in electrical measurements [52].

Reasons why this effect appears in some cases are unclear due to a lack of statistics and

spatiotemporal limitations of video acquisition. Nevertheless, one possible explanation is related

to reactor geometry where confining conditions are different in the two directions. The electrode

diameter is around 4 cm and glass boundaries are at approximately 3 cm of the electrode edges.


11

691 695 696

697 698 699

706 711 736

(a) 691 695 696

697 698 699

706 711 736

(b)

Figure 7. Heartbeat instability for a low repetition rate observed on the total

plasma glow: (a) some images in true colours during one contraction–expansion

sequence, (b) same images as in (a) with post-processing and false colours (from

dark blue to red).

Consequently, the plasma usually extends beyond the electrodes towards the lateral sides. The

plasma width is then nearly twice as large as the height and it is nearly the same for the trapped

dust cloud and the void. No strong electrostatic barrier exists due to the distance to the glass

boundaries, thus, plasma and void modifications in these directions are relatively free. On the

contrary, the electrodes imposed a strong electrostatic barrier and vertical modifications and

motions are then drastically controlled by the sheaths.

4. Plasma glow behaviour in the whole discharge

Plasma glow in the whole interelectrode volume is recorded during the instability in the two

different cases (low and high repetition rate). This analysis is performed in order to understand

global evolution of the plasma and not only central part behaviour as in the previous section.

It can identify some side effects already suggested (relation between a sharp peak in electrical

measurements and brighter glow regions near plasma horizontal boundaries [49, 51]) and that

are inaccessible through the analysis presented in section 3.

4.1. Sequences with a low repetition rate

The total plasma glow recorded by fast imaging during a contraction–expansion sequence is

shown in figure 7(a). Both bright presheath regions are clearly evidenced on the upper and lower

parts of the images (electrodes are not visible). Before the contraction–expansion sequence

(stable situation with an open void), the central plasma region is relatively uniform and not very

bright (image 691). Then, the glow increases fast and concentrates in the discharge centre before

disappearing and leaving a darker region in the centre as observed in the previous section. This

scheme is better evidenced by representing the same images in false colours and by subtracting

a reference image (stable situation) as shown in figure 7(b). The relatively homogeneous glow

(image 691) changes into a bright (i.e. higher ionisation) central region and dark (i.e. lower

ionisation) edges during the contraction (image 697). A reverse situation (dark centre and bright


12

Image number

Lin

e p

rofi

le (

pix

els)

100 200 300 400 500 600 700 800

50

100

150

200

250

300 (a)

Lin

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els)

620 640 660 680 700 720 740 760 7800

50

100

150200250300

Image number

Lin

e p

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685 690 695 700 705 710 7150

50100

150200

250

300

(b)

(c)

Figure 8. (a) Line profile (in false colours from dark blue to red) calculated from

the image series presented in figure 7; (b) and (c) are different magnifications

of (a).

edges) starts just before the re-opening (image 699). Then, the glow becomes slowly uniform

again (image 736).

Time evolution of a line profile is presented in figure 8(a) and magnified in figures 8(b)

and (c). To construct this false colour image, the horizontal line passing through the middle

of the discharge is extracted from each video image. This line has been chosen because it

passes through the region of biggest changes. In order to extract changes occurring during the

instability, a temporal average value is subtracted. A smoothing filter is also applied for noise

reduction and better delimitation of changing regions. Column profiling has also been performed

for the plasma glow analysis but no original information has been extracted partly due to the

limiting conditions imposed by electrodes.

Nearly isolated contraction–expansion sequences often have a short time duration. Figure 8

shows an instability frequency of about 3Hz. In between each sequence, original conditions

have time to be restored. From figures 8(b) and (c), plasma glow behaviour is well observed

and its complete evolution analysis is more easily performed than from direct images like in

figure 7. First, some small oscillations are observed before the real sequence. This phenomenon

has already been observed [51, 52] and corresponds to failed contractions appearing sometimes

near the instability existence threshold. This behaviour does not change the following analysis

and will be described in a future article. Figure 8 confirms that when the void contracts, the

plasma glow suddenly concentrates in the centre leaving dark border regions. The void re-

opening occurs during the reverse situation with two symmetrical bright regions surrounding

the plasma centre. Then, these regions seem to diffuse slowly to the centre until a homogeneous

glow is obtained once again.

In the previous section, only the central part of the plasma was observed when the

interference filter was removed. Indeed, the observed region was too small and due to the

viewing angle, out of focal plane information was lost. The whole plasma recording allows

to identify the dark (during contraction) and bright (during expansion) plasma edge regions.

These observations show that the increase of plasma glow in the centre is really a concentration

of the plasma towards the centre at the expense of the plasma edges. After that, the reverse

situation occurs with plasma edges brighter than in the stable void situation.


13

Image number

Lin

e pro

file

(pix

els)

685 695 705 71550

100

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Lin

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file (pix

els)

693 695 697 699 701100

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180

200

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260

280(a) (b)

a.u.

a.u.

0.385 0.39 0.395 0.4

Time (s)

a.u.




(c)

(d)

(e)

Figure 9. (a) Line profile (in false colours from dark blue to red) calculated

from the image series presented in figure 7 with electrical measurements super-

imposed; (b) zoom of (a); (c) electrical measurements; (d) central line (line 178)

profile of (a); (e) middle position line (line 105) profile of (a).

Correlation with electrical results is presented in figures 9(a) and (b) with non smoothed

images in order to get pixel precision. The dotted curve is the current continuous component.

From this superimposition it can be deduced that the contraction corresponds to an increase

in the global ionisation (i.e. electron density). Indeed, increase of central plasma glow

corresponds to an increase of electrical signal amplitude meaning that this central increase is

not compensated by the decrease in the near plasma edge. Then, when the reverse situation

occurs, ionisation appears to be below the stable situation value (as indicated by electrical

measurements). Indeed, the dark central region is not balanced by the larger but slightly

brighter near edge regions. Finally, these regions tend to rejoin in the centre following the slow

increase observed in electrical measurements. These behaviours are also visible by comparing

electrical measurements with line profiles (figures 9(c)–(e)) extracted from figure 9(a) (not to

be confused with temporal evolution of line profile extracted from video images) and physically

corresponding to the time evolution of individual pixels taken in (d) plasma centre and in (e) the

near edge.

4.2. Sequences with a high repetition rate

When the heartbeat instability frequency increases, contraction–expansion sequences are closer

in time, and no exact return to original stable conditions is achieved. Indeed, the next sequence

happens while the void size has not reached its stable state. The case presented here is

quite similar to the one shown in figures 5 and 6. The corresponding video is presented in

figure 10 showing three successive contraction–expansion sequences. Images extracted from

one sequence appear in figure 10(a). Figure 10(b) is obtained from figure 10(a) in the same way

as figure 7(b). As the sequence lasts longer, better time resolution is obtained and variation of

plasma glow is well evidenced. In comparison with the low repetition rate case, the change from

bright centre and dark edges to the reverse is here more strongly marked. The different phases of

the contraction–expansion sequence that have been already described, are highly visible here.

The deduced line profile is very significant and summarises nearly all results concerning

plasma glow evolution during the heartbeat instability (figure 11). Indeed, starting from a void


14

Figure 10. Heartbeat instability for a high repetition rate observed on the total

plasma glow: (a) some images in true colours during one contraction–expansion

sequence; (b) same images as in (a) with post-processing and false colours (from

dark blue to red) (see also movie 4, 2.7MB).

Image number

Lin

e pro

file

(pix

els)

50 100 150 200 250 300

50

100

150

200

250

300 (a)

Image number

Lin

e pro

file

(pix

els)

120 140 160 180 200 220 24060

80

100

120

140

160

180

200

220

240

260

280(b)

Figure 11. (a) Line profile (in false colours from dark blue to red) calculated from

image series presented in figure 10; (b) zoom of (a) with electrical measurements

superimposed.

expansion (dark inner part and bright plasma edges), slow concentration of the glow from the

plasma edge towards the centre takes place. This motion can be correlated to the grey corona

region observed in the dust cloud and presented in section 3. Furthermore, it clearly appears that

the two brighter edge regions do not meet completely and no homogeneous plasma is restored

as in figure 8. Indeed, the central glow increases at the expense of plasma edges before the

brighter glow regions fully converge towards the plasma centre. This central enhancement is

then followed by a strong and fast reversal. Electrical measurements are superimposed on the

line profile in figure 11(b). The bright region moving from the plasma edge towards the plasma

centre corresponds to a continuous increase of the current. Then, it appears that the strong

brightness reversal is related to the presence of a sharp peak in the current. Measurements

without peak certainly correspond to cases where the brightness reversal between plasma centre

and edges exists, but is not strongly marked (figure 9). This sharp peak is an interesting feature

already observed but not fully explained. Indeed, in [49], preliminary results suggested that

this peak could be correlated to a strong glow decrease in the centre and increase in the


15

a.u.

a.u.

a.u.

0.07 0.08 0.09 0.1 0.11 0.12 0.13

Time (s)

a.u.



Integrated central luminosity

(a)

(b)

(c)

(d)


a.u.

a.u.

a.u.

−0.03 −0.02 −0.01 0 0.01 0.02 0.03

Time (s)

a.u.


Central fibre

Intermediate fibre

Edge fibre

(e)

(f)

(g)

(h)

Figure 12. Extracted data from figure 11: (a) electrical measurements; (b) central

line profile; (c) line profile in the glow reversal region. (d) Plasma glow integrated

on a small central volume from complete image series of figure 10. Previous

measurements in similar conditions for (e) electrical measurements; (f) light

recorded by a centred optical fibre; (g) intermediate fibre; (h) edge fibre (same

observed region as (c)).

plasma edge. These measurements were performed thanks to a spatially resolved (four different

positions) analysis of an argon line emission at 750.4 nm. Further results with five optical fibres

(also used for characterisation of dust particle growth instabilities [53]) and integrating all

plasma wavelengths gave similar results [51] but no complete description was available. Present

results greatly enhance spatiotemporal resolution of this phenomenon and clearly correlate the

sharp peak with an enhancement of the plasma glow outside the central region. Variations of

electrical measurements are representative of global changes of the plasma (spatially integrated

measurement). From the present analysis it appears that electrical measurements are not only

dominated by the plasma core.

These effects are also evidenced in figure 12 by direct comparison of (a) electrical

measurements, with data extracted from figure 11: (b) central line profile (i.e. physically

corresponding to central pixel variation), (c) middle position line profile (i.e. variation of a pixel

in the glow reversal zone) and from the video in figure 10: (d) integrated central luminosity (i.e.

variation integrated in a small centred volume assumed to represent plasma core). Comparison

of figure 12(a), (b) and (d) shows that the main central plasma changes are observed in

electrical measurements. Nevertheless, it appears that the maximum value of central glow

occurs while electrical measurements have already decreased. In fact, at this time, electrical

measurements are affected by the edge plasma glow decrease as observed in figure 12(c).

Finally, the sharp peak region is well understood thanks to figure 11(b) and figure 12(a)–(d).

Opposite behaviour of electrical measurements and central glow is directly related to the strong

and fast enhancement of plasma edges while the centre becomes the less luminous part.

In order to show consistency of the new results, previous measurements performed

in similar instability conditions (i.e. similar electrical measurements) with the optical fibre

diagnostics, are presented in figure 12(e)–(h). Here, the edge fibre (figure 12(h)) observes the

region corresponding to the middle line profile of figure 12(c). The intermediate fibre observes


16

a region in between the centre and the edge. Its maximum value appearing before the one of

figure 12(f) traduces the plasma concentration towards centre. Main features concerning the

sharp peak region are observed: the peak corresponds to a glow reversal between the centre and

the edge. This comparison is of major interest because it confirms that no artefacts, neither in the

way measurements are performed nor in data retrieval from images, are implied for explanation

of the sharp peak region.

5. Discussion and conclusion

From the different analyses, several features of the heartbeat instability can be brought to

the fore: the void contraction corresponds to an increase of the central plasma glow. These

observations tend to show that a sudden higher ionisation in the plasma centre enhances the

ratio between the inward electrostatic force and the outward ion drag force. Then, the void centre

becomes darker but dust particles continue to move towards the centre due to their inertia. Their

motion seems more or less continuous (at relatively constant speed) and is stopped suddenly

(see for example figure 1). The void is not entirely filled by dust particles meaning that a

force prevents further motion towards its centre. This force also exceeds gravity that should

induce void collapse by acting on, among others, dust particles constituting the upper part of

the void. At this time the central glow is not very bright. Furthermore, it appears more or less

homogeneous, a property usually associated with a weak electric field. The force stopping dust

particles is then supposed to be the ion drag.

Dust particles located near the plasma centre are then nearly immobile while a dust particle

‘wave front’ coming from the plasma edge starts to be detected. At this time, a corona region of

higher ionisation is observed in plasma glow images. This region surrounds the plasma centre

and moves slowly towards it. Correlating dust cloud images with plasma glow ones, this moving

region seems to correspond to the dust particle wave front. Darker centre and brighter edge can

create an outward electrostatic force near the corona region. This hypothesis is assumed by

analogy with the contraction phase when an enhanced glow region attracts dust particles. Thus,

central dust particles are immobile (far from this interface) and the wave front corresponds to

dust particles close to the corona region and moving towards it. As dust particles move, local

conditions change and the corona region moves towards the centre attracting inner dust particles

one after the other. As the plasma becomes more and more homogeneous, the wave front speed

decreases and the last dust particles to move are finally the closest to the plasma centre.

In the case of low repetition rate sequences, the original stable position of the void is

restored between each sequence that can be considered as isolated and independent. On the

contrary, high repetition rate conditions lead to a continuous motion of the dust particles. Indeed,

while central dust particles return to their original position, a new contraction occurs.

An asymmetry in void contraction has been evidenced in some cases. This behaviour can

find its origin in the discharge geometry, confined by electrodes in the vertical direction and

more free to diffuse in the horizontal one. It is confirmed by the observed corona region in

plasma glow images. Indeed, it is better observed near the horizontal edges than near the vertical

ones where it is merged with the bright presheath regions.

The sharp peak in electrical measurements can be associated with the glow reversal from

bright centre and dark edges to bright edges and dark centre. It appears when this phenomenon

is strongly marked. It is replaced by a change in signal slope in other cases. This reversal is

very fast and our camera system does not allow to resolve this phenomenon in time. The optical


17

fibre diagnostics have a high time resolution but noise is relatively important and the data do

not show if there is a wave propagation between centre to edge during the reversal. The reversal

seems to be instantaneous and its physical origin is under investigation.

Acknowledgments

The PKE-Nefedov chamber has been made available by the Max-Planck-Institute for

Extraterrestrial Physics, Germany, under the funding of DLR/BMBF under grants No.

50WM9852. We would like to thank S Dozias for electronic support and J Mathias for optical

support. ESA and Kayser-Threde GmbH are acknowledged for providing the high speed camera

system in the framework of the IMPACT programme. This work was supported by CNES under

contract 02/CNES/4800000059.

References

[1] Walch B, Horanyi M and Robertson S 1994 IEEE Trans. Plasma Sci. 22 97

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[53] Mikikian M, Cavarroc M, Couëdel L and Boufendi L 2006 Phys. Plasmas 13 092103


Mixed-Mode Oscillations in Complex-Plasma Instabilities

Maxime Mikikian,* Marjorie Cavarroc, Lenaıc Couedel, Yves Tessier, and Laıfa Boufendi

GREMI, Groupe de Recherches sur l’Energetique des Milieux Ionises, UMR6606, CNRS/Universite d’Orleans,

14 rue d’Issoudun, BP6744, 45067 Orleans Cedex 2, France(Received 11 February 2008; published 6 June 2008)

Instabilities in dusty plasmas are frequent phenomena. We show that some instabilities can be described

by mixed-mode oscillations often encountered in chemical systems or neuronal dynamics and studied

through dynamical system theories. The time evolution of these instabilities is studied through the change

in the associated waveform. Frequency and interspike interval are analyzed and compared to results

obtained in other scientific fields concerned by mixed-mode oscillations.

DOI: 10.1103/PhysRevLett.100.225005 PACS numbers: 52.27.Lw, 05.45.!a, 52.25.Gj, 52.35.!g

Solid dust particles from a few nanometers to centi-

meters are often present in plasmas and are at the origin

of a wide variety of new phenomena. Plasmas containing

dust particles are called dusty or complex plasmas. These

media are encountered in many environments such as

astrophysics, industrial processes, and fusion devices.

Dust particles can come from regions in the plasma vicinity

or can be formed in the plasma due to the presence of

molecular precursors. In most cases, they acquire a nega-

tive charge by attaching plasma free electrons. In plasma

reactors, dust clouds are often characterized by a central

dust-free region. This region, usually named ‘‘void’’ [1–6],

is considered to be maintained by an equilibrium between

inward electrostatic and outward ion drag forces. Under

certain conditions, this equilibrium can be disturbed, re-

sulting in self-excited low frequency oscillations of the

void size. The obtained instability, consisting of successive

contractions and expansions of the void, is named ‘‘heart-

beat’’ [1,4,7,8] due to its apparent similarity with a beating

heart. Nevertheless, the dust cloud behavior is more com-

plex than it could be supposed [7,8]. This self-excited

instability can stop by its own through an ending phase

characterized by failed contractions that appear more and

more numerous as the end approaches. The phenomena

sustaining the instability evolve progressively until a new

stable state is reached. This phase can be studied by

analyzing the discharge current or the plasma glow emis-

sion. These signals have a well-defined shape that seems

similar to mixed-mode oscillations (MMOs).

MMOs are complex phenomena consisting of an alter-

nation of a varying number of small amplitude oscillations

in between two larger ones (also named spikes). Small and

large amplitude oscillations are often considered as, re-

spectively, subthreshold oscillations and relaxation mecha-

nisms of the system. These MMOs are encountered in a

wide variety of fields. In chemistry, reaction kinetics can

take the form of MMOs [9,10] like in the intensively

studied Belousov-Zhabotinskii reaction [11,12]. In natural

sciences, MMOs are the subject of an intense research [13–

15] since their observation in the Hodgkin-Huxley model

of neuronal activity [15,16]. In this context, they are

strongly related to spiking and bursting activities in neu-

rons [17,18]. In plasma physics, MMOs have been reported

in dc glow discharge [19,20] but to the best of our knowl-

edge, no evidence of MMOs in dusty plasmas have been

reported yet. Because of the broad diversity of scientific

domains concerned by MMOs, they induced a lot of re-

searches and several approaches have been explored. These

studies are based on dynamical system theories, like, for

example, canards [13,15,21], subcritical Hopf-homoclinic

bifurcation [16,22] or Shilnikov homoclinic orbits [23,24].

In this Letter, we report for the first time on the existence

of MMOs in dusty plasmas. An example is analyzed and its

main characteristics, like typical frequencies and evolution

of interspike interval, are explored. Particular attention is

paid to the evolution of the number of small amplitude

oscillations in between consecutive spikes. We thus under-

line very close similarities with typical MMOs observed in

neuronal activity and chemistry.

Experiments are performed in the PKE-Nefedov reactor

[2,25]. A capacitively coupled radiofrequency (rf at

13.56 MHz) discharge is created between two planar par-

allel electrodes. The static argon pressure is comprised

between 0.5 and 1.6 mbar and the applied rf power is

around 2.8 W. Dust particles are grown by sputtering a

polymer layer deposited on the electrodes and constituted

of previously injected melamine formaldehyde dust parti-

cles. A few tens of seconds after plasma ignition, a three-

dimensional dense cloud of visible grown dust particles

(size of a few hundreds of nanometers) is formed. This

observation is performed by laser light scattering with

standard CCD cameras. The discharge current (related to

electron density) is measured on the bottom electrode.

Plasma emission in the center is recorded thanks to an

optical fiber with a spatial resolution of a few millimeters

[8]. These two diagnostics give rather similar results and

are used to characterize the ending phase of the heartbeat

instability. When void contraction occurs, a high amplitude

peak is observed in the signals [8]. Failed void contractions

appear as small amplitude oscillations, i.e., failed peaks.

PRL 100, 225005 (2008)P H Y S I C A L R E V I E W L E T T E R S week ending

6 JUNE 2008

0031-9007=08=100(22)=225005(4) 225005-1 2008 The American Physical Society

The first two oscillations in Fig. 1(a) are representative of a

typical fully developed heartbeat signal [7] without any

failed peak. The observed shape is the one used as refer-

ence, with a main peak corresponding to the fast contrac-

tion and a slowly increasing signal (like a shoulder) during

the slow void reopening [8]. Then, a transition between this

regime and a regime with one failed contraction [arrow in

Fig. 1(a)] occurs. At the instant where a large amplitude

oscillation is expected, a decrease in the signal occurs. The

spike occurrence is then delayed. A similar behavior is

obtained in the optical signal, as observed in Fig. 1(b) for a

transition between one and two failed peak regimes. Signal

shape clearly corresponds to typical MMOs and is remark-

ably similar to waveforms measured or modeled in chemi-

cal systems [9,10,12,24] or in neuronal dynamics

[13,15,18,26,27]. For the sake of simplicity, classical no-

tation for MMOs [9,12] will be used in the following. Thus,

a pattern characterized by L high amplitude oscillations

and S small amplitude ones will be noted LS. As an

example, Fig. 1(a) is a transition between 10 and 11 states,

whereas Fig. 1(b) corresponds to a 11–12 transition. A

closer look to this last transition is shown in Fig. 1(c) where

the first 12 state is superimposed on the last 11 state. A

vertical dotted line marks the place where signals differ. As

it can be observed, signals are nearly identical until this

time and no clear indication announces this next change. In

dynamical systems, noise can induce drastic changes in

system behavior. In the present study, due to a low signal to

noise ratio, analyses cannot be performed on raw data.

Thus, data in Fig. 1 have been processed using wavelet

techniques (checking that no signal distortion was added)

in order to remove noise and thus its effect will not be

investigated.

As the similarity between our signals and MMOs is

established, a more complete study can be performed.

For that purpose, electrical signal evolution has been re-

corded until the complete end of the instability. A typical

sequence is shown in Fig. 2(a). The signal mean value is

almost constant and the instability consists of a succession

of high amplitude spikes separated by an interspike interval

(ISI) increasing as one goes along. As in Fig. 1, low

amplitude oscillations appear in between these spikes giv-

ing the classical shape of MMOs. The increase in the ISI is

thus a consequence of the occurrence of more and more

failed peaks. As an example, the insert in Fig. 2(a) shows a

14–15 transition (this time series starts during the 13 state).

Just after the transition, the memory that a transition oc-

curred is visible. Indeed, the new failed peak [Fig. 2(a)

inset] has a higher amplitude than the other ones, keeping

partly the characteristics of a spike. Then, its amplitude

decreases during the next few patterns until reaching the

same amplitude as the other failed peaks. These MMOs

can be also characterized by their corresponding phase

space. Figures 2(b) and 2(c) are, respectively, 3D and 2D

phase spaces of a representative part of the time series

presented in Fig. 2(a). To obtain these attractors, an appro-

priate time delay ! has been calculated using the mutual

information method proposed in [28] in the framework of

the Belousov-Zhabotinskii reaction. The main large trajec-

tory in phase space corresponds to the large amplitude

variation during spikes. A smaller loop (indicated by an

arrow labeled 1) corresponds to a short additional peak

sometimes observed on the right part of the main spikes

[7,8]. This phenomenon is observed in Fig. 1(a) and does

not alter the present study. Small amplitude oscillations are

represented by loops in a tiny region indicated by an arrow

labeled 2 [Figs. 2(b) and 2(c)]. In this region, the number of

loops corresponds to the number of failed peaks (3 loops

for the 13 state, for example) like observed in [15]. The

analysis through ISIs is often used in neuroscience for

0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27

0.566

0.568

0.57

0.572

Ele

ctri

cal

sig

na

l (a

.u.)

0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

−0.5

0

0.5

Op

tica

l

sig

nal

(a.u

.)

0.28 0.29 0.3 0.31 0.32 0.33

−0.5

0

0.5

Time (s)O

pti

cal

sig

na

l (a

.u.)

(a)

(b)

(c)

1 2

1 2

1 1

1 1

FIG. 1 (color online). (a) Electrical signal with a transition

between 10 and 11 states (arrow: failed peak). (b) Optical signal

with a transition between 11 and 12 states. (c) Superimposition of

the two consecutive states shown in (b).

0 2 4 6 8 10 12 14 16 18 20

0.668

0.67

0.672

Time (s)

Ele

ctri

cal

sign

al

(a.u

.)

0.6680.67

0.6720.665 0.67 0.675

0.665

0.67

0.675

X(t)X(t+τ)

X(t

+2τ)

0.666 0.668 0.67 0.672 0.6740.666

0.668

0.67

0.672

0.674

X(t)

X(t+

τ)

1 2

2

1

(a)

(c) (b)

FIG. 2 (color online). (a) Evolution of the electrical signal

during the instability ending phase. The insert is a zoom of a

transition between 14 and 15 states. Corresponding phase space

is represented in (b) 3D and (c) 2D. Arrows labeled 1 and 2 show,

respectively, the loop due to the additional peak observed on the

right-hand side of spikes [see Fig. 1(a)] and the region containing

the loops corresponding to failed peaks.


6 JUNE 2008

225005-2

characterizing neural activity [18,29]. Figure 2(a) shows

that the ISI increases by the progressive addition of failed

peaks until the instability stops. This behavior is better

observed by performing a spectrogram, i.e., a time-

resolved fast Fourier transform (FFT) of this time series

[Fig. 3(a)]. This analysis clearly shows a step-by-step

decrease of the characteristic frequencies until no more

oscillations are observed (at "17:7 s). The main frequency

corresponding to the spike occurrence is indicated by an

arrow labeled 1. Because of the shape of the time series,

harmonics are also obtained. Figure 3(a) clearly shows that

the frequency jumps become smaller as time increases.

Moreover, the duration of a given LS state decreases with

time. These last two properties are also brought to the fore

by performing a global FFT of the time series [Fig. 3(b)].

Clearly separated peaks are observed, confirming the step-

by-step evolution of the frequency. The main peaks are

located at 20.5, 17.5, 15.1, 13.3, and 11.8 Hz. Their ampli-

tude is proportional to the duration of each LS state. As in

Fig. 3(a), it is observed that this duration decreases with

time (as the frequency). The amplitude of the first peak at

20.5 Hz is not representative because the time series started

to be recorded during the 20.5 Hz phase (13 state) and part

of this phase is thus missing. Another correlation that can

be made with Fig. 3(a), is the decrease of the frequency

jumps. Indeed, the frequency jumps are equal to !f # 3,

2.4, 1.8, and 1.5 Hz. Finally, Fig. 3(a) gives also the

frequency of the failed peaks (arrow labeled 2). It is

interesting to note that this frequency does not strongly

evolve but only slightly decreases with time (variation of

about 10%).

In order to improve the analysis of ISI, the occurrence

time of each spike is deduced from the time series pre-

sented in Fig. 2(a). Spike position is obtained by detecting

points exceeding a correctly chosen threshold. The direct

time evolution of the ISI can thus be obtained as shown in

Fig. 4(a) (blue dots). A very clear step-by-step behavior, as

in [14,16], is obtained with ISIs varying from "0:05 s

("20:5 Hz, 13 state) to "0:4 s ("2:5 Hz, 140 state). Each

step corresponds to the occurrence of a new failed peak,

i.e., transition from a LS to a LS$1 state. Close to the end of

the instability (after 15 s), the system evolves faster and

faster with transitions between LS and LS$n states with nincreasing from 2 to 5. The global time dependence (with-

out taking into account the steps) of ISI evolves as a$b=%t0 ! t& as in [16,22]. Parameters a, b and t0 are fitted to

the experimental data and the corresponding curve is

superimposed in Fig. 4(a) (orange curve). The parameter

t0 corresponds to the time of the asymptotic limit ("19 s)

where ISI tends to infinity which is often considered to be

characteristic of a homoclinic transition [16,30]. Fig-

ure 4(a) also shows that the ISI during a LS state is not

perfectly constant and slightly increases (left hand side

insert). This increase is nonlinear (like, for example, in

[16]) and seems to become faster close to the transition

between LS and LS$1 states. Thus, it appears that this

transition is the consequence of a progressive increase of

the ISI. It can be noted that this ISI increase during a given

LS state can also be fitted by a function a$ b=%t0 ! t&(blue curve in left hand side insert) with different a, b, and

t0 than the ones obtained from the fit of the global curve.

This behavior was not so clearly marked in Fig. 3(a) due to

the smoothing induced by the time window used for cal-

culating the time resolved FFT. Nevertheless, a slight

decrease in frequency during each LS state can be observed

on the high order harmonics. As in Fig. 3(b), the duration

of each LS state can also be analyzed by performing a

Time (s)

Fre

qu

ency

(H

z)

0 2 4 6 8 10 12 14 16 18

20

40

60

80

100

120

140

160

2 4 6 8 10 12 14 16 18 20 220

0.5

1

x 10−3

Frequency (Hz)

Po

wer

sp

ectr

um

2

1

(a)

(b)

FIG. 3 (color online). (a) Spectrogram of the time series pre-

sented in Fig. 2(a). Arrows labeled 1 and 2 show the occurrence

frequency of, respectively, spikes and small amplitude oscilla-

tions. (b) Power spectral density of the same signal.

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (s)

ISI

(s)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

20

40

ISI (s)

Occ

ure

nce

nu

mb

er

(a)

(b)

FIG. 4 (color online). (a) Evolution of the ISI during the

instability ending phase: measurements (blue dots) and corre-

sponding fit (orange curve). The two inserts are zooms showing,

Left: slight evolution of ISI during a LS phase, Right: unstable

transition between two phases. (b) Corresponding histogram of

ISIs with an exponential fit (green curve).


6 JUNE 2008

225005-3

histogram of ISIs [Fig. 4(b)]. This histogram is character-

ized by a clear multimodal pattern that can be fitted by a

decreasing exponential function [green curve, first bar is

not taken into account due to missing data as in Fig. 3(b)].

Same conclusions can be drawn: when ISI increases, du-

ration of the corresponding state decreases.

This ISI analysis reveals also a particular behavior

which is the unstable transition between a LS and a LS$1

state [right-hand side inset in Fig. 4(a)]. The system

changes from a LS state to a LS$1 one but returns to the

LS state once again before changing of state definitively.

Experimental observations, not presented here, show that

an alternation of two states differing by one failed peak can

be observed during several periods. Thus, transitions can

take place through the progressive occurrence of LS$1

patterns during a LS state. It suggests that the step-by-

step evolution of Fig. 4(a) could be compared to the devil’s

staircase which is an infinite self-similar staircase struc-

ture. This specific structure has been observed in chemistry

in the framework of MMOs [9,10,12,31] but also in a wide

range of other fields like in physiology [32] or in wave-

particle interaction [33]. In chemical systems, this structure

is often obtained by representing the firing number F #L=%L$ S& as a function of the control parameter. The

number of steps is then nearly infinite and in between

two parent states, an intermediate state exists with a firing

number related to the ones of the parent states by the Farey

arithmetic [9,12,31]. Thus, the structure obtained in

Fig. 4(a) can be an incomplete devil’s staircase with an

unidentified control parameter evolving with time and not

perfectly flat plateaus. More intermediate states could be

obtained with a more slowly varying system and thus it can

be speculated that the staircase could be partly completed.

In this Letter, we evidenced and characterized mixed-

mode oscillations in dusty plasmas. This work highlights

new situations of MMOs that can be of interest for improv-

ing dynamical system theories related to these structures.

The obtained structures are very similar to what is observed

in oscillating chemical systems and in neuronal activity.

These fields use well-known sets of equations giving rise to

MMOs. This scientific background can thus be used to

explore and develop new theoretical approaches in dusty

plasma dynamics.

M. Lefranc is acknowledged for helpful discussions. The

PKE-Nefedov chamber has been made available by the

Max-Planck-Institute for Extraterrestrial Physics,

Germany, under the funding of DLR/BMBF under Grants

No. 50WM9852. This work was supported by CNES under

Contract No. 02/CNES/4800000059.

*[email protected]

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[24] M. T. M. Koper, Physica (Amsterdam) 80D, 72 (1995).

[25] A. P. Nefedov, G. E. Morfill, V. E. Fortov, H. M. Thomas,

H. Rothermel, T. Hagl, A. Ivlev, M. Zuzic, B. A. Klumov,

and A. M. Lipaev et al., New J. Phys. 5, 33 (2003).

[26] G. S. Medvedev and J. E. Cisternas, Physica (Amsterdam)

194D, 333 (2004).

[27] M. A. Zaks, X. Sailer, L. Schimansky-Geier, and A. B.

Neiman, Chaos 15, 026117 (2005).

[28] A. M. Fraser and H. L. Swinney, Phys. Rev. A 33, 1134

(1986).

[29] G. S. Medvedev, Phys. Rev. Lett. 97, 048102 (2006).

[30] U. Feudel, A. Neiman, X. Pei, W. Wojtenek, H. Braun, M.

Huber, and F. Moss, Chaos 10, 231 (2000).

[31] K.-R. Kim, K. J. Shin, and D. J. Lee, J. Chem. Phys. 121,

2664 (2004).

[32] S. De Brouwer, D. H. Edwards, and T. M. Griffith, Am. J.

Physiol. Heart Circ. Physiol. 274, 1315 (1998).

[33] A. Macor, F. Doveil, and Y. Elskens, Phys. Rev. Lett. 95,

264102 (2005).


6 JUNE 2008

225005-4

D.3 APPENDIX D. PUBLICATIONS


• Couedel, L.; Mikikian, M.; Boufendi, L. & Samarian, A. A., “Residual dust charges in

discharge afterglow Phys. Rev. E, 74, 026403 (2006)

• Couedel, L.; Samarian, A.; Mikikian, M.& L.Boufendi, “Dust-Cloud Dynamics in a com-

plex plasma afterglow”, IEEE Trans. Plasma Sci., In press (2008)

• Couedel, L.; Samarian, A.; Mikikian, M. & L.Boufendi, “Influence of the ambipolar-to-

free diffusion transition on dust particle charge in a complex plasma afterglow”, Phys.

Plasmas, 15, 063705 (2008)

• Couedel, L.; Samarian, A.; Mikikian, M. & L.Boufendi, “Dust density influence on complex

plasma decay”, Phys. Lett A, In press (2008)

215

Residual dust charges in discharge afterglow

L. Couëdel,* M. Mikikian, and L. BoufendiGREMI (Groupe de Recherches sur l’Énergétique des Milieux Ionisés), CNRS/Université d’Orléans, 14 rue d’Issoudun,

45067 Orléans Cedex 2, France

A. A. SamarianSchool of Physics A28, University of Sydney, NSW 2006, Australia

!Received 24 May 2006; published 22 August 2006"

An on-ground measurement of dust-particle residual charges in the afterglow of a dusty plasma was per-formed in a rf discharge. An upward thermophoretic force was used to balance the gravitational force. It wasfound that positively charged, negatively charged, and neutral dust particles coexisted for more than 1 minafter the discharge was switched off. The mean residual charge for 200-nm-radius particles was measured. Thedust particle mean charge is about −5e at a pressure of 1.2 mbar and about −3e at a pressure of 0.4 mbar.

DOI: 10.1103/PhysRevE.74.026403 PACS number!s": 52.27.Lw

I. INTRODUCTION

Dusty or complex plasmas are partially ionized gas com-posed of neutral species, ions, electrons, and charged dustparticles. In laboratory experiments, these particles can beeither injected or grown directly in the plasma. Injected dustparticles are usually micron-size particles. Due to their mass,they are confined near the bottom electrode where the elec-tric force counterbalances gravity. The microgravity condi-tion is necessary to study dust clouds of micrometer-sizeparticles filling the whole plasma chamber #1$. In the labo-ratory, dense clouds of submicron particles light enough tocompletely fill the gap between the electrodes can be ob-tained using reactive gases such as silane #2,3$ or using atarget sputtered with ions from plasma #4–7$.

Dust-particle charge is a key parameter in complexplasma. It determines the interaction between a dust particleand electrons, ions, its neighboring dust particles, and elec-tric field. The determination of the dust-particle charge isthus one of the basic problems in any complex plasma ex-periments. Knowledge of dust charge will allow us to under-stand the basic properties of dusty plasma, particle dynamicsin dust clouds, and methods to manipulate the particles.

The particle charge controls the dust dynamics both inlaboratory and technological plasma reactors, and also inspace plasmas. Thus one of the main dusty plasma chal-lenges is to understand the dust charging in a wide range ofexperimental conditions, which simulates industrial andspace plasmas. For example, in most industrial processes inthe microelectronics industry which uses silane as the reac-tive gas, dust contamination is a vital problem #8$. The dustdynamics and coagulation in space plasma are also phenom-ena governed by dust-particle charges #9,10$.

There are many publications reporting on the investiga-tion of dust charging in discharge plasma; see #11–18$ andreferences therein. However there are only a few papers de-voted to dust charging, or decharging to be more specific, inthe discharge afterglow. In #19$ a diffusion of fine dust par-

ticles !8–50 nm" in the afterglow of a dusty plasma has beenstudied. From the afterglow decay of dust number density thecharged fraction of particles has been measured. It was foundthat some particles carried a very small residual charge andsome were neutral. A model has been proposed to explaincharges on particles in the late afterglow of a dusty plasma.An observation of decay, or “decharging,” of an rf plasmawith dust particles after switching off the discharge powerwas reported in #20$. Experiments were performed in thePKE-Nefedov reactor #1$ under microgravity conditions onboard the International Space Station. The existence of nega-tive residual charges has been shown. A simple theoreticalmodel that describes data obtained was proposed. The re-sidual charges were attributed to the presence of an excita-tion electric field that was used for dust charge measure-ments. The limitations of the Space Station experimentroutine could not prove this hypothesis. Theoretical predic-tions on an effect of rf plasma parameters on residual chargeshas also not been verified.

So any new experimental evidence of residual dustcharges in afterglow plasma will lead to a better understand-ing in the decharging of complex plasma. Indeed the valueand nature of residual charges after plasma extinction is ofgreat importance. The dust-particle charge in afterglowplasma could induce a problem in future single-electron de-vices where a residual charge attached on deposited nano-crystals would be the origin of dysfunction. It could makeeasier industrial plasma processing reactor decontaminationthanks to the use of specially designed electric fields. Theresidual charges on dust particles in fusion reactors !such asITER" also can make the cleaning process much easier.

In this paper, we report an on-ground experiment on theresidual charge measurement of dust particles after the decayof a dusty plasma. The experiment was performed in thePKE-Nefedov reactor where the dust particles were physi-cally grown in the plasma. A temperature gradient was intro-duced in the chamber to create an upward thermophoreticforce to balance gravity. The residual charges were deter-mined from an analysis of dust oscillations, which were ex-cited by applying a sinusoidal bias to the bottom electrode. Acoexistence of positively and negatively charged dust par-*Electronic address: [email protected]

PHYSICAL REVIEW E 74, 026403 !2006"

1539-3755/2006/74!2"/026403!9" ©2006 The American Physical Society026403-1

ticles as well as noncharged ones was found for more than1 min after the discharge was switched off. The residualcharges for 200-nm-radius particles have been measured fortwo different pressures. It was revealed that dust particleskept the residual charges only when the discharge wasabruptly switched off. In the case when the discharge poweris decreased slowly until the plasma disappeared, there wasno residual charge on the dust particles. It was also shownthat the presence of the low-frequency excitation electricfield did not play any role in keeping the charge on dustparticles in the afterglow plasma.

The article is organized as follows. Section II is devotedto a general description of the experimental setup. In Sec. IIIwe analyze the forces acting on dust particles, paying mostattention to the thermophoretic force. In Sec. IV we discuss aprocedure of residual charge measurement and present ex-perimental data obtained, and in Sec. V we analyze discharg-ing of dust particle in afterglow plasma. Section VI is theconclusion.

II. EXPERIMENTAL SETUP

The work presented here is performed in the PKE-Nefedov !Plasma Kristall Experiment" chamber designed formicrogravity experiments #1$. It is an rf discharge operatingin push-pull excitation mode. It consists of 4-cm-diam par-allel electrodes separated by 3 cm. The injected power variesin the range 0–4 W. Dust particles are grown in an argonplasma !0.2–2 mbar" from a sputtered polymer layer depos-ited on the electrodes and coming from previously injecteddust particles !3.4 !m, melamine formaldehyde". A detaileddescription of this experiment and previous results are pre-sented in Refs. #1,4,5$.

For the study concerning residual charges, the top elec-trode was cooled !Fig. 1". An upward thermophoretic forcewas applied to dust particles in order to counterbalance grav-ity #21$ when the plasma is off. To study particle charges, asinusoidal voltage produced by a function generator withamplitude ±30 V and frequency of 1 Hz was applied to thebottom electrode. The induced low-frequency sinusoidalelectric field E!r , t" generated dust oscillations if they kept aresidual electric charge.

A thin laser sheet perpendicular to the electrodes illumi-nates dust particles and the scattered light is recorded at 90°with standard charge coupled device !CCD" cameras with 25images per second. Video signals were transferred to a com-

puter via a frame-grabber card with 8-bit gray scale and560"700 pixel resolution. In order to avoid edge effect, afield of view over 8.53"5.50 mm2 restrained to the center ofthe chamber was used for residual charge measurement. Bysuperimposition of video frames particle trajectories havebeen obtained. The coordinates of the particles were mea-sured in each third frame. The amplitude of the oscillationswas figured out from the measured particle positions. Abso-lute values for the oscillation amplitude were obtained byscaling the picture pixels to the known size of the field ofview.

III. FORCES ACTING ON DUST PARTICLES

Dust particles in laboratory plasmas are subjected to vari-ous forces that confine them in the plasma or drag them out!to the walls or pump outlet" #8$: the confining electrostaticforce, the gravity, the thermophoretic force, the ion dragforce, and the neutral drag force.

After the discharge is switched off, the forces still actingon dust particles are the gravity Fg, the thermophoretic forceFT, the neutral drag force Fdn, and, if dust particles keepresidual charges, the electric force FE due to the electric fieldinduced by the sinusoidal voltage applied to the lower elec-trode. In this experiment the thermophoretic force balancedthe gravity force and the electrostatic force was the reasonfor the dust oscillations while the neutral drag force dampensthe last ones.

The expression for the gravity force is

Fg =4

3#rd

3$g , !1"

where g is the gravity acceleration, rd the dust particle ra-dius, and $ its mass density.

The neutral drag force was taken as #22$

Fdn = −8

3%2#rd

2mnnnvTn&1 + %ac#

8'!vd − vn" , !2"

where mn is the neutral atom mass, nn the neutral atom den-sity, vTn=%8kBTn /#mn the thermal speed with Tn=T the neu-tral gas temperature, kB the Boltzmann constant, %ac the ac-commodation coefficient, vd dust particle speed, and vn amean speed of neutral atoms. In our experiment vn can betaken as 0 because there is no gas flow.

The electric force can be written as

FE!r,t" = QdE!r,t" , !3"

where Qd is the dust particle electric charge and E!r , t" is theelectric field between the electrodes after the plasma isswitched off. In the late afterglow the plasma density is verysmall and E!r , t" can be approximated by the vacuum fieldabove a charged disk. Taking into account the size of thecamera field of view, the size of the electrode, and the factthat camera was directed to the region near center of thechamber, we are interested in the electric field vertical com-ponent E!r , t"(Ez!z , t"e!z)E!z , t"e!z only. The calculatedvalue of E!z , t" at the central axis is presented in Fig. 2.

The expression for the thermophoretic force must be cho-sen carefully. Indeed, it depends strongly on the Knudsen

ELECTRODE

ELECTRODE

COOLING SYSTEM

GL

AS

S

GL

AS

S

plasma & particleslasersheet

field ofview

FIG. 1. !Color online" Schematic of the experimentalapparatus.

COUËDEL et al. PHYSICAL REVIEW E 74, 026403 !2006"

026403-2

number Kn= l /rd #23$ where l is the mean free path of buffergas species. In our experiment, we work at an operating pres-sure around 1 mbar. In a previous paper #4$, the size ofgrown dust particles was reported between 200 nm and800 nm. It gives, using results from Varney #24$ for an atom-atom cross section, Knudsen number 250&Kn&1000. Con-sequently we operate in a free molecular regime where a dustparticle is similar to a very large molecule. Many theorieshave been developed #23,25–28$ and used #21,23,29$ forthermophoresis in the free molecular regime. The most com-monly used equation is the Waldmann equation #28$ whichhas been verified experimentally #30,31$:

FT = −32

15rd

2 ktr

vTn! T , !4"

where !T is the temperature gradient in the gas and ktr thetranslational part of the thermal conductivity given for amonoatomic gas by #32$

ktr =15kB

4mn!ref& T

Tref''

, !5"

where !ref is the reference viscosity at reference temperatureTref =273 K and the exponent ' results from a best fit ofexperimental viscosity near the reference temperature. Forargon, !ref =2.117"10−5 Pa s and '=0.81 #32$.

Another important effect that must be considered in theestimation of thermophoretic force is the influence of a finitevolume of gas. If the pressure is low enough, the gas meanfree path can become comparable to a length scale of experi-mental apparatus and the gas can no longer be treated as acontinuous medium. Under such a condition, an additionalKnudsen number must be added #23$, KnL= l /L, where L isthe length scale of the reactor. In this experiment, the lengthbetween electrodes is L=3 cm giving KnL*5"10−3 whichmeans the gas can be considered as a continuous medium.

The temperature gradient between the electrodes was cal-culated with FEMLAB !steady-state analysis of heat transferthrough convection and conduction with heat flux, convec-tive and temperature boundary conditions using theLagrange-Quadratic element". The temperatures on the elec-trodes were measured by a thermocouple and used as bound-ary conditions for the problem. The contour plot obtained is

presented in Fig. 3. It shows that the vertical component ofthe temperature gradient is constant near the reactor center.The value of temperature gradient is about 2 K/cm for ourexperiment. There is also a small horizontal component ofthe temperature gradient. This is the reason for particle driftin the horizontal direction. Such a drift allowed us to resolveparticle trajectories !see next section" and made particlecharge measurement more convenient.

IV. EXPERIMENTAL PROCEDURE AND RESULTS

In this section, dust-particle size estimation and residualcharge measurement are presented. The residual charge mea-surements have been performed by the following routine.First the chamber was pumped down to lowest possible pres-sure !base pressure *2"10−6 mbar" and the cooling systemwas turned on. After this, argon was injected up to the oper-ating pressure, the discharge was started, and particles weregrown, forming familiar structures such as a void !see, forexample, #5$". Then, the discharge was switched off and thebottom electrode was biased by sinusoidal voltage.

In afterglow plasma, the dynamics of dust particles is de-termined by a temperature gradient and excitation electricfield. Figure 4 presents a superimposition of images takenafter discharge had been switched off. There are two differ-ent types of motion observable. Dust particles drift upwards,downwards, and to the side due to existing temperature gra-dients and they oscillate due to electrostatic force. It is obvi-ous that the thermophoretic force acts on any dust particle inthe chamber, while the electrostatic force acts only if par-ticles have charge in afterglow. Thus the presence of oscil-lating particles !see Fig. 4" clearly indicates that dust par-ticles do have residual charges after the discharge has beenswitched off. Dust particles oscillating in opposite phases aswell as nonoscillating dust grains have been observed, indi-cating that negatively charged, positively charged, and non-charged dust particles coexist after plasma extinction. It isworth mentioning that in order to observe dust oscillationsthe discharge must be switched off abruptly. It was shownthat if the power was decreased slowly until the plasma dis-

550

600

650

700

750

800

850

900

0.9 1 1.1 1.2 1.3 1.4 1.5

Ele

ctri

c fi

eld

(V/m

)

Distance to the lower electrode (cm)

FIG. 2. !Color online" Vertical component of the electric field inthe camera field of view.

FIG. 3. Temperature profile and gradient.

RESIDUAL DUST CHARGES IN DISCHARGE AFTERGLOW PHYSICAL REVIEW E 74, 026403 !2006"

026403-3

appears, there are no residual charges !no oscillations of dustparticles in the sinusoidal electric field were observed". An-other interesting fact is that the residual charge on dust par-ticles has a long relaxation time and does not depend on timewhen the excitation electric field was applied. Dust oscilla-tions were observed for more than 1 min after plasma extinc-tion and in both cases when the function generator wasswitched on during the discharge or a few seconds after thedischarge is turned off.

As can be seen in Fig. 4, there are dust particles fallingafter the discharge is switched off. These particles are too bigto be sustained by the thermophoretic force. Other particlesare horizontally adrift at constant height; this means that forthese particles the gravity force is balanced by the thermo-phoretic force. These particles have been used to measureresidual charges. It is clear from Fig. 4 that use of a large-field-of-view camera gives us a nice overall picture of decay-ing dusty plasma but it is not suitable for residual chargemeasurement because edge effects cannot be neglected. Thusa camera with a small field of view was used for the chargemeasurement !see Fig. 4". The superimposition of imagesfrom this camera is presented in Fig. 5. These images give usa clear track of the dust oscillations, so dust grain trajectoriescan be reconstructed !Fig. 6".

A. Size and mass of levitating dust grains in discharge

afterglow

Particle size !mass" and residual charge measurement arestrongly related in this experiment. Charge, size, and mass ofdust particles have to be determined. Considering that dustparticles levitating in a reactor at a constant height afterplasma extinction are the ones for which the gravity is ex-actly balanced by the thermophoretic force, the dust particleradius can be found using Eqs. !1" and !4":

rd = −8

5#$g

ktr

vth! T . !6"

Dust grains are supposed to be spherical and mainly made ofcarbon !sputtering of a carbonaceous polymer material" #4$

so that the mass md can be deduced from Eq. !6":

md =4

3#rd

3$ , !7"

where $ is the mass density of graphite. In our experimentalcondition, the radius of levitating dust grains is estimated tord+190 nm and their mass is md+6.5"10−17 kg.

B. Dust-charge measurement

From the measurement of oscillation amplitude, the re-sidual charge on a dust particle can be obtained. As the grav-ity is compensated by the thermophoretic force, the equationof motion for one dust particle, neglecting its interactionswith other dust particles, can be reduced to

mdz = FE!z,t" + Fnd!z" . !8"

Taking E!t"=E0!zmean" cos!(t" !the amplitude of the electricfield E0 is the one at the mean dust levitation height zmean"and using Eqs. !3", !2", and !8", the dust-particle oscillationamplitude b can be obtained #33$:

UPPER ELECTRODE

CHARGE MEASUREMENT

USED FOR RESIDUAL

OF THE CAMERA

2

1

FIELD OF VIEW

FALLING DUST PARTICLES

DUST PARTICLES OSCILLATING WITH

THE EDGE ELECTRIC FIELD

FIG. 4. Superimposition of video frames taken with a large-field-of-view CCD camera. Arrows 1 and 2 represent, respectively,the vertical and horizontal components of the temperature gradient.Edge effects as well as falling dust particles can be seen.

FIG. 5. Superimposition of video frames 10 s after plasma ex-tinction. Dust-particle oscillations can clearly be seen. The tempera-ture gradient has a slight horizontal component. Therefore, oscilla-tions are in the two-dimensional laser plane.

0 0.5 1 1.5 2 2.5−60

−40

−20

0

20

40

60

time (s)

Z−

po

siti

on

(p

ixel

s)

grain1grain 2grain 3grain 4grain 5grain 6

FIG. 6. !Color online" Oscillation of six dust grains 10 s afterplasma extinction. Nonoscillating dust grains and opposite phaseoscillations are observed.


026403-4

b!(,Qd,E0" =QdE0!zmean"

md(%(2 + 4)2/md2

, !9"

where )= !4/3"%2#rd2mnnnvTn#1+%ac!# /8"$ is the damping

coefficient and (=2#f where f is the frequency imposed bythe function generator. Equation !9" can easily be inversed toderive the dust residual charge:

Qdres=

mdb„(,Qd,E0!zmean"…(%(2 + 4)2/md2

E0!zmean". !10"

The sign of the dust-particle charge is deduced from thephase of the dust-particle oscillation with respect to the ex-citation electric field. Oscillation amplitudes up to 1.1 mmhave been measured !depending on the operating pressure"and charges from −12e to +2e are deduced where e is theelementary charge. It has been found that at high pressuredust particles keep a higher mean residual charge !Table I"but uncertainties are of *2e for each measurement.

V. DISCUSSION

The charging !discharging" process of dust particles in aplasma is governed by the contributions of all currents enter-ing !or leaving" the dust surface, involving plasma electronsand ion currents, photoemission and thermionic emissioncurrents, etc.

dQd

dt= , Ia + , Il, !11"

where Ia and Il are currents absorbed and emitted by theparticle !with appropriate sign". In most cases for dischargeplasmas we can ignore the emission current and the kineticsof the particle charge can be expressed as

dQd

dt= Ji − Je = − #erd

2#nevTee−* − nivTi

!1 + Te*"$ , !12"

where Je and Ji are the fluxes of electrons and ions onto theparticle, vTi!e"=%8kBTi!e" /#mi!e" the thermal velocity of ions

!electrons", Te=Te /Ti is the electron to ion temperature ratio,ni!e" is the density of ions !electrons", and *=−eQd /4#+0kBrdTe is the dimensionless surface potential of adust particle where +0 is the vacuum dielectric permittivity.According to Eq. !12", charge on a dust particle depends on

the electron-ion masses, temperatures, and density ratiosme /mi, ne /ni, and Te /Ti. Thus to analyze the decharging ofdust particle in afterglow plasma one needs to consider thekinetics of plasma decay.

The plasma diffusion loss and electron temperature relax-ation determine the kinetics of the discharge plasma decay#34$. In the presence of dust particles, plasma loss is due todiffusion onto the walls completed by surface recombinationon dust particles. The equations for the plasma density andelectron temperature relaxation are #34,35$

dn

dt= −

n

,L, !13"

dTe

dt= −

Te − 1

,T, !14"

where n=ni,e /n0 with n0 the initial plasma density, ,L thetime scale of the plasma loss, and ,T the time scale for elec-tron temperature relaxation. The expressions for the timescales are #34,35$

,L−1 = ,D

−1 + ,A−1, !15"

,D−1 +

linvTi

3-2 !1 + Te" )1

2!1 + Te"

1

,D.

, !16"

,A−1 + #rd

2ndvTi!1 + *Te" ) &1 + *Te

1 + *' 1

,A.

, !17"

,T−1 =%#

2%me

mi

vTi

len

%Te )%Te

,T.

, !18"

where ,D is the ambipolar diffusion time scale onto the walls,,A the particle absorption time scale, li!e"n the mean free pathof the ion- !electron-" neutral collision, and - the character-istic diffusion length !-*1 cm in this experiment". The .exponent stands for the limit at very long time.

For a charging time scale lower than the plasma decay ortemperature relaxation time scales the charge on the dustparticle is in equilibrium—i.e., ion and electron fluxes bal-ance each other—*+*eq, and using Eq. !12", *eq is given by

ne

ni

%Tee−*eq =%me

mi!1 + Te*eq" . !19"

In this case, the expressions for the charge fluctuation andcharge fluctuation time scale ,Q are

dQd

dt+ −

Qd − Qdeq

,Q, !20"

,Q−1 +

vTird

4/i02 !1 + *eq"n )

n

,Q0 , !21"

where /i0=%+0kBTi /n0e2 is the initial ion Debye length.It should be noted that the time scale for dust charge

fluctuations strongly depends on the plasma density and can

TABLE I. Measured maximum, minimum, and mean dust-particle residual charges for the two operating pressures.

P=1.2 mbar P=0.4 mbar

!T −190 K m−1 −177 K m−1

rd 194 nm 180 nm

md 6.9"10−17 kg 5.4"10−17 kg

) 1.36"10−13 kg s−1 0.39"10−13 kg s−1

Qdresmax +2e +2e

Qdresmin −12e −4e

Qdresmean −5e −3e


026403-5

vary from microseconds for the initial stages of plasma de-cay up to seconds in the case of almost extinguished plasma.Taking into account Eqs. !13" and !21", the time dependenceof ,Q can be expressed as #20$

,Q−1 =

1

,Q0 exp!− t/,L" . !22"

To understand the dusty plasma decharging dynamics wehave to compare different time scales. In Table II, the timescales for this experiment are presented. It can be seen thatthe initial charge fluctuation time scale is the shortest. Thetemperature relaxation time scale is shorter or becomes com-parable !for 0.4 mbar" to the plasma density decay time scaleand plasma losses mainly determined by diffusion. The latestmeans that for our experimental conditions dust particles didnot affect the plasma decay at the initial stage. Figure 7presents the qualitative dependence of the main plasma and

dust particle parameters during the afterglow. Four stages ofthe dust plasma decay can be labeled. As we can see, the firststage of the plasma decay !t&,T" is characterized by theelectron temperature Te drop down to room temperature,while the plasma density !especially in case ,T

0 &,L0" is

slightly decreased. As the charging time scale is almost in-

dependent of Te #Eq. !21"$, the charge is still determined byits equilibrium value #Eq. !12"$. During the temperature re-laxation stage the particle charge should decrease to thevalue #20$

QrT =1

Te0& *eq!1"

*eq!Te0"'Q0 +

Q0

62+ − 15e , !23"

where Q0 is the initial dust charge in the plasma and QrT thevalue of the dust residual charge at the end of the first decaystage. The dust charge in the plasma Q0 was estimated asQ0=−950e, solving numerically Eq. !19", with the given pa-rameters Ti=300 K and Te+3 eV, for argon plasma withni-ne. At the next stage of decay, the electron temperature isstabilized while the plasma density is still decreasing !seeFig. 7". So ,Q continues increasing according to Eqs. !12"and !21". When ,Q becomes comparable to ,L, the particlecharge cannot be considered as in equilibrium and to deter-mine particle charge we should use Eq. !12". The time scalewhen the particle charge starts sufficiently deviating from theequilibrium can be estimated as #Eqs. !15" and !22"$

td * − ,L. ln.8

3&/i0

-'2 lin

rd/ * 6,L

.. !24"

However, according to Eq. !12", as long as the plasma isneutral !ne=ni" the charge on the dust particle does notchange. The plasma will keep quasineutrality until decayingrates for the electrons and ions are the same. It will be true inthe case of ambipolar diffusion. When the nature of the dif-fusion changes, electrons and ions start to diffuse indepen-dently, which will lead to a changing ne /ni ratio and conse-quently the dust charge change.

The nature of the plasma diffusion changed when the par-ticle volume charge cannot be ignored or when the plasmascreening length becomes comparable to the chamber size. Inthe first case the ion diffusion will be influenced by the nega-tively charged dust particles, while the electrons will be freeto go. In the second case, large density differences appearover distances less than the screening length and electronsand ions diffuse independently. Let us estimate the charac-teristic times for both cases.

The influence of the overall particle charge is determinedby the value of the Havnes parameter Pe=−NQd /ene. Basedon the model discussed, the qualitative evolution of theHavnes parameter Pe in dusty plasma afterglow can beplotted as shown in Fig. 8. The initial value of Pe is small!*0.06 with an estimated dust density N*2"105 cm−3 andn0+ne0*5"109 cm−3" and there is no influence of dust. Atthe first stage of decaying !temperature relaxation stage" Pedecreases due to a dramatic decrease of the dust charge whilethe plasma density decreases by a factor of 1.1. At ,T, Pereaches its minimum value of about 1"10−3. After this Pestarts increasing. During this stage the dust particle charge

TABLE II. Values of the different time scales for the two oper-ating pressures.


,D0 30 !s 90 !s

,D. 1.5 ms 5 ms

,A0 0.4 ms 0.4 ms

,A. 30 ms 30 ms

,L0 30 !s 90 !s

,L. 1.4 ms 4.3 ms

,T0 100 !s 34 !s

,T. 1 ms 340 !s

,Q0 4 !s 4 !s

tc 20 ms 60 ms

,Q!tc" 1.4 s 1.4 s

Qd

n /i

ne

n0 Te / T

in /

in

eTe / T

i

n0

Qd

t ptct

d

1 1

tT

IVIIII II

Time (s)Plasmaoff

FIG. 7. !Color online" Qualitative time evolution of dust charge,plasma density, and electron temperature during the afterglow. Fourstages of the dust plasma decay can be identified: I: temperaturerelaxation stage up to tT. II: plasma density decay stage up tp. III:dust charge volume stage tc. IV: frozen stage.


026403-6

changes slowly while the plasma number density decays fast!see Fig. 7". The time at which Pe becomes *1 can be esti-mated as #Eqs. !13"–!18"$

tp * ,L. ln.&Te0

Tn'&− en0

QdN'/ * 8,L

.. !25"

The screening length becomes comparable to the chambersize—i.e., /i!nc"*-—when the density drops down tonc=/i0

2 /-2. This occurs at #20$

tc * ,L. ln nc

−1 * 10,L.. !26"

At time min#tp , tc$, electrons start running away faster thanions and the ratio ni /ne grows. For our experimental condi-tions tp& tc; thus, the neutrality violation due to the presenceof dust particles happens before the Debye length exceedsthe chamber size. So the third stage of dusty plasma decaystarts at tp. During this stage the charge on the dust particlesis changed due to the changing of the ne /ni ratio. At thisstage td& tp, thus, the kinetic equation !12" should be usedfor estimations of the charge variation. The upper limit of thecharge change can be estimated ignoring the electron currentand considering the time interval between tp and tc,

0dQ

dt& Ji0

tp

tc!27"

&#erd2ni!tp"vTi

0&1 −e

4#+0kBrdTiQd'0

tp

tc. !28"

Solving Eq. !28", the charge should evolve as follows:

Qd = &QdT −1

%'exp!− K%0t" +

1

%, !29"

where %=e /4#+0kBrdTi*0.28/e, K=#erd2ni!tp"vTi

*190e,and thus !K%"−1*20 ms*0t= !tc− tp".

Therefore, the charge during the third stage decreases to−3e. At the fourth stage of plasma decay, t1 tc, the plasmadensity decreases such that any further changes of the dustcharge become negligible and the charge remains constantfor a while. Thus the final residual charge for our condition is

expected to be about Qdres*−3e which is well correlated

with the charges measured in the experiment !Table I".The observation of neutral and positively charged par-

ticles in our experiments can be explained by the particlecharge distribution. The charge distribution in a dust-particleensemble is due to charge fluctuation on every individualdust particle. It has been shown #36–40$ that the rms ofstochastic charge fluctuations varies as 2%1Qd2, where 2 is aparameter depending on the plasma conditions and close to0.5. Thus for mean charge around −3e the charge distributionshould lie mainly between −1e and −5e. However, in ourcase we can expect a broader distribution, taking into ac-count two reasons. First for small particles with a charge ofabout a few electrons the image force !polarizability" correc-tion starts playing role; see #37$. And for electron tempera-ture less then 0.1 eV the correction factor can be up to 3.That is a significantly broadened distribution, so dust par-ticles can experience fluctuations to neutral and positivecharges; see Fig. 9. Another reason for the broadened distri-bution could be the earlier freezing of the charge distributionso it remains the same while the mean charge decreases.Indeed as t1 td the charge is not in equilibrium with thesurrounding plasma, so the existing charge distributionshould remain unchanged or changed slightly during the laterstages. Unfortunately, in this experiment the ratio betweenpositively and negatively charged particles has not been mea-sured due to experimental limitations concerned with the factthat dust grains do not stay in the laser sheet for a sufficientlylong time due to their out-of-plane thermal motion. Thus toelaborate on the true reason for the presence of residualcharges of different polarities one needs additional measure-ments of the charge distributions.

The considered decharging model predicts a higher re-sidual charge for lower pressure. For low pressure the tem-perature relaxation time scale exceeds the plasma densitydecay time !see characteristic times for 0.4 mbar in Table II".This leads to a quick increase of the charging time andmakes charge freezing at an early stage of decay keepingcharge at a relatively high value. In fact larger charges weremeasured for the higher pressure. A possible explanation for

P0

Pmin

Pe

t p

1

I II III

Time (s)

tT

FIG. 8. Qualitative time evolution of the Havnes parameter afterthe power is switched off.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

−2 0 2 4 6

Ch

arg

e d

istr

ibutio

n

−Q/e

FIG. 9. Charge distribution with and without image forcecorrection. The dotted line represents the distribution with0Q=0.5%Q. The solid line represents the distribution with0Q=0.87%Q #37$.


026403-7

this is an electron reheating in late afterglow from metastableatoms #41$. The presence of a significant amount of argonmetastables in PKE discharge have been established byplasma modeling #42$ and by experimental measurements#43$. Due to reheating, the electrons can have temperatures afew times more than room temperature which will lead to anincrease of the residual charge on dust particles. As reheatingis more effective for high pressures, larger charges are thusexpected.

VI. CONCLUSION

Residual dust-particle charges have been measured in thelate afterglow of a dusty plasma. Dust particles have beengrown directly in the plasma by sputtering of a polymer ma-terial previously deposited on electrodes. Dust particles witha radius of few hundreds of nanometers are levitating afterthe power of the discharge is switched off. The gravity wasbalanced by an upward thermophoretic force. Residualcharges were determined from an analysis of dust oscilla-tions, which were excited by applying a sinusoidal bias to thebottom electrode.

Positive, negative, and noncharged dust particles havebeen detected. The mean residual charge for 200-nm-radiusparticles was measured. The particle charge is about −5e at apressure of 1.2 mbar and about −3e at a pressure of0.4 mbar. A model for the dusty plasma decay was exploitedto explain the experimental data. According to this model thedusty plasma decay occurs in four stages: temperature relax-

ation stage, density decay stage, dust-charge volume stage,and frozen stage !ice age IV". The main decrease of the dustcharge happens during the first stage due to cooling of theelectron gas. The final residual charge established during thethird stage when the density of ions exceeds the density ofelectrons and the plasma density is still high enough to affectthe charge. Measured values of the dust residual charges arein good agreement with values predicted by the model. How-ever, the residual charge dependence on discharge conditionsand the detection of positively charged particles show that amore detailed model taking into account various phenomena!electron reheating, electron release #44$, afterglow chemis-try" in decaying plasma has to be developed for a betterunderstanding of dusty plasma afterglow.

ACKNOWLEDGMENTS

The authors would like to thank S. Dozias and B. Dumaxfor electronic support, J. Mathias for optical support, andY. Tessier for experimental support. The PKE-Nefedovchamber has been made available by the Max-Planck-Institute for Extraterrestrial Physics, Germany, under thefunding of DLR/BMBF under Grant No. 50WM9852. Thiswork was supported by CNES under Contract No. 793/2000/CNES/8344. This work is partly supported by the French-Australian integrated research program !FAST" of the Frenchforeign affairs ministry and the International Science Link-ages established under the Australian Government’s innova-tion statement, Backing Australia’s Ability and University ofSydney Research and Development Scheme.

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Morfill, A. P. Nefedov, V. E. Fortov, and the PKE-Nefedovteam, New J. Phys. 5, 19.1 !2003".

#5$ M. Mikikian and L. Boufendi, Phys. Plasmas 11, 3733 !2004".#6$ D. Samsonov and J. Goree, J. Vac. Sci. Technol. A 17, 2835

!1999".#7$ A. A. Samarian and B. W. James, Phys. Lett. A 287, 125

!2001".#8$ A. Bouchoule, Dusty Plasmas: Physics, Chemistry and Tech-

nological Impacts in Plasma Processing !Wiley, New York,1999".

#9$ I. A. Belov, A. S. Ivanov, D. A. Ivanov, A. F. Pal, A. N.Starostin, A. V. Filippov, A. V. Demyanov, and Y. V. Petrush-evich, JETP 90, 93 !2000".

#10$ U. Konopka et al., New J. Phys. 7, 227 !2005".#11$ S. Vladimirov, K. Ostrikov, and A. Samarian, Physics and

Applications of Complex Plasmas !Imperial Press, London,2005".

#12$ E. B. Tomme, D. A. Law, B. M. Annaratone, and J. E. Allen,Phys. Rev. Lett. 85, 2518 !2000".

#13$ C. Zafiu, A. Melzer, and A. Piel, Phys. Rev. E 63, 066403!2001".

#14$ A. A. Samarian and S. V. Vladimirov, Phys. Rev. E 67,066404 !2003".

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Thomas, Phys. Rev. Lett. 89, 175001 !2002".#22$ P. Shukla and A. Mamun, Introduction to Dusty Plasma Phys-

ics !IOP, Bristol, 2002".#23$ F. Zheng, Adv. Colloid Interface Sci. 97, 255 !2002".#24$ R. N. Varney, Phys. Rev. 88, 362 !1952".#25$ Z. Li and H. Wang, Phys. Rev. E 70, 021205 !2004".#26$ I. Goldhirsch and D. Ronis, Phys. Rev. A 27, 1616 !1983".#27$ I. Goldhirsch and D. Ronis, Phys. Rev. A 27, 1635 !1983".#28$ L. Waldmann, Z. Naturforsch. A 14, 589 !1959".#29$ K. DeBleecker, A. Bogaerts, and W. Goedheer, Phys. Rev. E


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71, 066405 !2005".#30$ W. Li and E. J. Davis, J. Aerosol Sci. 26, 1063 !1995".#31$ W. Li and E. J. Davis, J. Aerosol Sci. 26, 1085 !1995".#32$ M. A. Gallis, D. J. Rader, and J. R. Torczynski, Aerosol Sci.

Technol. 36, 1099 !2002".#33$ L. Landau and E. Lifchitz, Physique Théorique !Édition Mir

Moscou, 1982", Vol. 1.#34$ Y. P. Raizer, Gas Disharge Physics !Springer, Berlin, 1991".#35$ V. N. Tsytovich, Phys. Usp. 40, 53 !1997".#36$ T. Matsoukas and M. Russell, J. Appl. Phys. 77, 4285 !1995".#37$ T. Matsoukas, M. Russell, and M. Smith, J. Vac. Sci. Technol.

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026403-9


IEEE TRANSACTIONS ON PLASMA SCIENCE 1

Dust-Cloud Dynamics in a ComplexPlasma Afterglow

Lénaïc Couëdel, Alexandre A. Samarian, Maxime Mikikian, and Laïfa Boufendi

Abstract—The motion of a dust-particle cloud in the afterglowof a complex plasma is reported. It is shown that the dust-particlecharges strongly affect the dynamic behavior of the dust cloud.Consequently, using the analysis of the whole cloud motion todeduce dust residual-charge and plasma-diffusion parameters isproposed.

Index Terms—Charge, dust cloud, dusty plasma.

ACOMPLEX (dusty) plasma is a partially ionized gas in

which there are some charged dust particles. They are

very common media and can be either natural plasmas (such as

comet tails, planetary ring, and near-Earth plasmas) or human-

made plasmas (such as tokamak-edge plasma, etching plasmas,

and plasma-enhanced chemical vapor deposition). In laboratory

discharges, the dust particles are negatively charged due to a

higher mobility of the electrons [1], [2].

The dust particles can be either grown or injected directly

into the plasma. Injected dust particles are generally microme-

ter sized, and thus, they are confined in the sheath region above

the lower electrode where the force due to the electric field can

balance the gravity force. Consequently, microgravity condi-

tions are needed to observe a cloud of injected particles filling

up the whole discharge chamber [3]. On the other hand, dust

particles grown directly in the plasma are light enough to fill the

whole interelectrode space, and dense clouds of dust particles

can be observed and can form structures, such as the dust void,

in onground experiments [4]. Dust particles can be grown by us-

ing reactive gases (such as silane [5] or methane [6]) or by sput-

tering a target with the ions coming from the plasma [7], [8].

The dust-particle charge is a very important parameter for

complex plasmas. It determines the interaction of a dust particle

with its neighboring dust particles and the ions and electrons of

the surrounding plasma. Consequently, the determination of the

dust-particle charge is one of the most important measurements

in any complex plasma experiment.

Manuscript received November 28, 2007; revised February 13, 2008. Thiswork was supported in part by CNES under contract 793/2000/CNES/8344 andin part by the French–Australian integrated research program (FAST) underContract FR060169.

L. Couëdel is with the School of Physics, The University of Sydney,Sydney, N.S.W. 2006, Australia, and also with the Groupe de Recherchessur l’Energétique des Millieux Ionisés, Centre National de la RechercheScientifique–Université d’Orléans, 45067 Orléans Cedex 2, France.

A. A. Samarian is with the School of Physics, The University of Sydney,Sydney, N.S.W. 2006, Australia.

M. Mikikian and L. Boufendi are with the Groupe de Recherchessur l’Energétique des Millieux Ionisés, Centre National de la RechercheScientifique–Université d’Orléans, 45067 Orléans Cedex 2, France.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2008.920220

In a decaying complex plasma, the electrons and ions of the

plasma are lost by diffusion to the walls of the reactor and

by recombination on the dust-particle surfaces. Consequently,

the charge carried by the dust particles evolves with the other

plasma parameters during the plasma afterglow. As the dust-

charging time depends strongly on the electron and ion den-

sities [1], the dust-particle charging time becomes very long

in the late plasma afterglow, and the dust-particle charges are

considered frozen. It has thus been shown that in the late

afterglow of a complex plasma, dust particles indeed keep a

residual charge of a few elementary charges and that positively

charged, negatively charged, and neutral dust particles coexist

for several minutes after the power of the discharge is turned off

[9], [10].

In a complex plasma afterglow, the dust-cloud motion will

be affected by the dust residual charges due to the involved

electrostatic interactions. By recording the motion of a dust

cloud in the afterglow of a complex plasma, it is thus possible

to deduce some important parameters such as the dust-particle

mean charge and the remanent electric fields. In this paper,

the motion of a dust-particle cloud in the afterglow of an RF

discharge is reported.

The experiment has been done in the plasma kristal ex-

periment (PKE)-Nefedov reactor [3]. It is an RF discharge

operating in a push–pull excitation mode. The electrodes are

parallel and are 4 cm in diameter. They are separated by a gap

of 3 cm. The injected power is in the range of 0–4 W. The

dust particles were grown in an argon plasma (0.2–2 mbar).

A thin laser sheet that is perpendicular to the electrodes was

used to illuminate the dust particles, and the scattered light was

recorded at an angle Θ ∼30 with a fast charge-coupled device

camera with 500 frames/s.

Frames from the video taken at different times of the complex

plasma afterglow are shown in Fig. 1. As it can be seen when

the discharge is on (t < 0), the interelectrode space is filled

with a cloud of dust particles in which a dust-free region called

a “void” can be observed. This void is a common feature of

complex plasmas [1]–[4]. The intensity of the scattered light of

the column going through the center of the void is also shown

in Fig. 1 (3-D plot). It shows its evolution with time and, thus,

the evolution of the dust density. A similar method was used in

[4] to study dust-cloud dynamics in a running discharge. The

void is clearly observed because it corresponds to a strong drop

in the scattered light intensity. Video frames can also be used to

extract dust-particle trajectories (not presented here).

When the discharge is turned off, the dust cloud starts to

fall down due to the gravity (in Fig. 1, frames for t > 0).

Nevertheless, it can be seen that in the first 20 ms after the

0093-3813/$25.00 © 2008 IEEE


2 IEEE TRANSACTIONS ON PLASMA SCIENCE

Fig. 1. (Top) Frames extracted from the video of the laser scattered light during the plasma afterglow. (Bottom) Evolution of the scattered intensity of the centralcolumn going through the center of the dust void. The zero coordinate in the vertical position corresponds to the top of the picture. The discharge was switchedoff at t = 0 ms.

discharge is switched off, the scattered light intensity increases

drastically. As the scattered light is roughly proportional to

the dust density, it indicates that the dust cloud contracts at

the very beginning of the plasma decay. The time scale of

this contraction (tc ≤ 20 ms) is too short to be explained by

the gravity force. Consequently, it implies the action of the

confining electric field. This electric field confines the dust

cloud inside the plasma when the discharge is on while the dust

particles tend to repel each other. When the RF generator is

turned off, the RF field decreases, and the confining electric

field evolves, too. During this time, the electron temperature

relaxes to room temperature, leading to a drastic drop of the

dust charge [9]. The dust-particle cloud thus contracts due to

the changes in the repelling and confining forces. Knowing

the dust-particle charge evolution and looking at the dust-

cloud dynamics, an estimation of the evolution of the confining

electric field may be done.

Another feature of the dust-cloud dynamics in the plasma

afterglow is the evolution of the dust void. As it is shown in

Fig. 1, the dust void remains well defined with a sharp boundary

for a very long time.

To conclude, the analysis of the motion of the dust cloud

could give information about dust residual charges, diffusion

processes, and electric fields because the dust-cloud dynamics

is strongly affected by the dust-particle charges.

ACKNOWLEDGMENT

The authors would like to thank S. Dozias and B. Dumax

for the electronic support, J. Mathias for the optical support,

and Y. Tessier for the experimental support. The PKE-

Nefedov chamber has been made available by the Max-

Planck-Institut für extraterrestrische Physik, Germany, under

the funding of the Deutsches Zentrum für Luft- und

Raumfahrt/Bundesministerium für Bildung und Forschung

under Grant 50WM9852.

REFERENCES

[1] S. Vladimirov, K. Ostrikov, and A. Samarian, Physics and Applications

of Complex Plasmas. London, U.K.: Imperial Press, 2005.[2] A. Bouchoule, Dusty Plasmas: Physics, Chemistry and Technological

Impacts in Plasma Processing. New York: Wiley, 1999.[3] A. P. Nefedov, G. E. Morfill, V. E. Fortov, and H. M. Thomas et al., “PKE-

Nefedov: Plasma crystal experiments on the international space station,”New J. Phys., vol. 5, p. 33, 2003.

[4] M. Mikikian, L. Couëdel, M. Cavarroc, Y. Tessier, and L. Boufendi, “Self-excited void instability in dusty plasmas: Plasma and dust cloud dynamicsduring the heartbeat instability,” New J. Phys., vol. 9, p. 268, 2007.

[5] M. Cavarroc, M. Mikikian, G. Perrier, and L. Boufendi, “Single-crystalsilicon nanoparticles: An instability to check their synthesis,” Appl. Phys.Lett., vol. 89, no. 1, p. 013 107, Jul. 2006.

[6] H. T. Do, G. Thieme, M. Fröhlich, H. Kersten, and R. Hippler, “Ion mole-cule and dust particle formation in Ar/CH4, Ar/C2H2 and Ar/C3H6 radio-frequency plasmas,” Contrib. Plasma Phys., vol. 45, no. 5/6, pp. 378–384,Aug. 2005.

[7] M. Mikikian et al., “Formation and behaviour of dust particle clouds ina radio-frequency discharge: Results in the laboratory and under micro-gravity conditions,” New J. Phys., vol. 5, p. 19, 2003.

[8] D. Samsonov and J. Goree, “Particle growth in a sputtering discharge,”J. Vac. Sci. Technol. A, Vac. Surf. Films, vol. 17, no. 5, pp. 2835–2840,Sep. 1999.

[9] L. Couëdel, M. Mikikian, L. Boufendi, and A. A. Samarian, “Residualdust charges in discharge afterglow,” Phys. Rev. E, Stat. Phys. Plasmas

Fluids Relat. Interdiscip. Top., vol. 74, no. 2, p. 026 403, Aug. 2006.[10] A. Ivlev et al., “Decharging of complex plasmas: First kinetic observa-

tions,” Phys. Rev. Lett., vol. 90, no. 5, p. 055 003, Feb. 2003.

Influence of the ambipolar-to-free diffusion transition on dust particlecharge in a complex plasma afterglow

L. Couëdel,1,2,a! A. A. Samarian,1 M. Mikikian,2 and L. Boufendi21School of Physics A28, The University of Sydney, Sydney, NSW 2006, Australia2GREMI (Groupe de Recherches sur l’Énergétique des Milieux Ionisés), CNRS/Université d’Orléans,14 rue d’Issoudun, 45067 Orléans Cedex 2, France

!Received 13 February 2008; accepted 12 May 2008; published online 27 June 2008"

The influence of diffusive losses on residual dust charge in a complex plasma afterglow has been

investigated. The residual charge distribution was measured and exhibits a mean value Qdres

#!−3e−5e" with a tail in the positive region. The experimental results have been compared with

simulated charge distributions. The dust residual charges were simulated based on a model

developed to describe complex plasma decay. The experimental and simulated data show that the

transition from ambipolar to free diffusion in the decaying plasma plays a significant role in

determining the residual dust particle charges. The presence of positively charged dust particles is

explained by a broadening of the charge distribution function in the afterglow plasma. © 2008American Institute of Physics. $DOI: 10.1063/1.2938387%

I. INTRODUCTION

Dusty plasmas !also called complex plasmas" are ionized

gases containing dust particles. These dust particles are elec-

trically charged due to interaction with ions and electrons of

the surrounding medium. In laboratory experiments, these

particles can be either injected or grown directly in the

plasma. Injected dust particles are usually micrometer-size

particles. Due to their mass, they are confined in the sheath

region where the electric force counterbalances gravity. Mi-

crogravity conditions are thus necessary to study dust clouds

of micrometer-size particles filling the whole plasma

chamber.1

In the laboratory, dense clouds of submicrometer

particles light enough to completely fill the gap between the

electrodes can be obtained using reactive gases such as

silane2,3

or using a target sputtered with ions from the

plasma.4–9

The dust particles are also subject to other forces

in the plasma such as the ion drag force, the neutral drag

force, and the thermophoretic force.10,11

Dust particle charge is a key parameter in a complex

plasma. It determines the interaction between a dust particle

and electrons, ions, its neighboring dust particles, and elec-

tric field.11,10

Data of the dust charge will allow us to under-

stand the particle dynamics in dust clouds, and methods of

manipulating the particles. The knowledge of the dust charge

is very important in every dusty medium !complex plasmas,

colloidal suspensions, fog". For example, in gas flow con-

taining dust particles the dust charge and dust charge distri-

bution are studied because of their consequences !safety haz-

ard, dry coating, etc.";12,13the charging of fog particles is

also studied as the electrical properties of fog are very im-

portant for the fog forecast and control; and aerosol and dust

systems influence the electric fields and currents in the

atmosphere.14

There are many publications reporting on the investiga-

tion of dust charging in a discharge plasma !see Refs. 10 and

15–22 and references therein". However, there are only a few

papers devoted to dust charging, or discharging to be more

specific, in the discharge afterglow.23–25

These papers report

that dust particles retain residual electric charges when the

power of the discharge is switched off. Nevertheless, the

discharging phenomenon was not totally understood. The

models proposed do not predict accurately residual charge

values observed in those experiments, nor do they explain

the existence of positively charge particles.

One of the main reasons is that the models are working

with simple assumptions. For example, it was assumed24

that

ambipolar diffusion continues until the ionic Debye length

!Di reaches the diffusion length ", and then an abrupt tran-

sition to free diffusion of ions and electrons occurs. In con-

trast, it has been shown that diffusion of charged species in a

plasma afterglow deviates from ambipolar diffusion as soon

as the ratio !" /!De"#100, where !De is the electron Debye

length.26–29

These results indicate that electrons and ions are

lost at different rates very early in the decay process and one

has to expect that this will strongly affect the value of re-

sidual dust charge. As the charge on dust particles is related

to the ratio ni /ne, where ni!e" is the ion !electron" density,10

this must result in a different dust particle charge evolution

in the afterglow compared to the predictions based on previ-

ous assumptions.24,25

For this reason, a model taking into

account the actual transition from ambipolar to free diffusion

of electrons and ions is vital for the correct calculation of

dust charges in afterglow plasma.

In this paper we report numerical simulations of residual

electric charge on dust particles in an afterglow plasma for

various transitions from ambipolar to free diffusion. These

numerical results are compared to the experimental data. The

importance of the transition from ambipolar to free diffusion

in the decaying plasma for determining the dust particle re-

sidual charges is shown. The broadening of the charge dis-

tribution function is found to be responsible for the presence

of positively charged particles.a"Electronic mail: [email protected].

PHYSICS OF PLASMAS 15, 063705 !2008"

1070-664X/2008/15"6!/063705/11/$23.00 © 2008 American Institute of Physics15, 063705-1

Downloaded 27 Jun 2008 to 129.78.64.102. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp

II. MODELING OF DISCHARGE AFTERGLOW

In this section, we first describe a qualitative model that

helps us to explain the existence of residual charges. Then

we consider dust-free plasma decay taking into account the

experimental transition from ambipolar to free diffusion. We

then discuss the plasma decay in the presence of dust par-

ticles using a model for particle charging !discharging" in a

afterglow plasma.

A. Four-stage model

The evolution of dust charge in afterglow plasma will

depend on the evolution of plasma parameters. In a decaying

plasma !afterglow plasma", the kinetics of plasma losses is

mainly governed by the electron temperature relaxation pro-

cess and plasma diffusion and recombination processes.30

The proposed model in Ref. 25 predicts evolution of dust

residual charge after the discharge is switched off. Based on

the plasma decay kinetics we can define four stages in a

discharge afterglow during which the different processes

!electron temperature relaxation, ambipolar diffusion, free

diffusion" govern the evolution of dust charge:

!1" Electron temperature relaxation stage: the electron tem-

perature decreases to the gas temperature while the

plasma density remains almost constant resulting in a

strong decrease of dust particle charges !t#500 #s for

an argon pressure PAr#1.2 mbar".!2" Ambipolar diffusion stage: the dust particle charge stays

almost constant while plasma particles !ions and elec-

trons" are lost by ambipolar diffusion on to the walls of

the reactor. The ambipolar diffusion of ions and elec-

trons continues until the Debye length !D equals the

diffusion length " of the reactor or until the Havnes

parameter PH= &Qd¯ nd /ene & #1 !t#40 ms for an argon

pressure PAr#1.2 mbar".!3" Free diffusion stage: this stage begins when diffusion

changes from ambipolar to free; i.e., electrons and ions

start diffusing independently. As electrons diffuse faster

than ions, the ratio ni /ne becomes more than 1, resulting

in a decrease of the dust particle charge !t#60 ms for an

argon pressure PAr#1.2 mbar".!4" Finally, ion and electron densities become too small to

influence dust particle charges that can be considered as

frozen !i.e., charging time tends to infinity".

This model predicts the existence of a negative residual

charge of a few electrons but is unable to predict a positive

residual charge. The limits of the model are the step transi-

tion from ambipolar to free diffusion and the failure to prop-

erly take account of the influence of dust particles. Indeed,

the effect of the dust particles is ignored until the dust charge

density is the same as the electron density, which leads to an

abrupt change from ambipolar to free diffusion of ions and

electrons. The assumption of the step transition from ambi-

polar to free diffusion is quite unphysical and contradicts

existing experimental data.26–29

These data indicate that elec-

trons and ions are diffusing at different rates very early in the

decay process. This smooth transition must lead to a change

in dust charges as the ratio ni /ne will be modified early in the

decaying process. Under the conditions of our experiments,

the ratio !" /!De"#50 when the discharge is switched off.

Consequently, this must result in a different dust particle

charge evolution in the afterglow compared to previous

assumptions.24,25

For this reason, in the next sections, a

modified model taking into account the gradual transition

from ambipolar to free diffusion is used to simulation of

complex plasma afterglow.

B. Dust-free plasma decay

In a dust-free plasma of monoatomic gas, losses of the

plasma are mainly due to diffusion to the walls. At the very

beginning of the afterglow when the discharge is switched

off, plasma density n as follows,

dn

dt= −

n

$D, !1"

where n=n /n0 with n0 the initial plasma density and $D is

the time scale for ambipolar diffusion given by30

$D−1 =

Da

"2'

linvTi

3"2!1 + Te" (

1

2!1 + Te"

1

$D% , !2"

where Da is the ambipolar diffusion coefficient, Te=Te /Ti

with Te the temperature of electrons, Ti=T the temperature of

ions and T the temperature of neutral atoms, vTi

=)8kBT /&mi the thermal speed of ions with kB the Boltz-

mann constant and mi the ion mass and lin=1 / !nn'in" the

mean free path for ion-neutral collisions, nn the density of

neutral atom, and 'in the ion-neutral cross section.

Nevertheless, it has been shown that the plasma diffu-

sion deviates from ambipolar diffusion as soon as the ratio

!" /!De"#100.26–29

From this moment, ions and electrons

must be treated separately. In our model, the deviation from

ambipolar diffusion is derived from the results of Gerber and

Gerardo29

or Freiberg and Weaver.26

Both experiments were

performed in helium at different pressures exhibit similar

evolution of the diffusion coefficients. Though these results

were obtained with helium, they can be used to estimate the

transition in an argon plasma. Indeed, the ambipolar diffu-

sion coefficient is

Da (!kBTi"

3/2

P'inmi1/2

, !3"

where P is the neutral gas pressure. As 'inAr+ #2.5'inHe+

!Ref. 31" and miAr+ #10miHe+, DaAr#!1 /8"DaHe

for equal

pressures. Consequently, it can be assumed that the diffusion

of an argon plasma with argon pressure PAr is the same as a

helium plasma with helium pressure PHe'8PAr. The experi-

mental results obtained for helium are in the range of pres-

sure 0.4–4 Torr !Ref. 26" or 9–22.8 Torr,29

which corre-

spond to 0.05–0.5 Torr and 1.13–2.85 Torr, respectively, in

equivalent argon pressure. Consequently, these sets of data

can be used for upper and lower estimation in the simula-

tions that are performed in the range 0.3–1 Torr

!0.4–1.3 mbar".In the model, electron and ion densities are treated sepa-

rately, and follow

063705-2 Couëdel et al. Phys. Plasmas 15, 063705 "2008!


dni

dt= −

ni

$Di, !4"

dne

dt= −

ne

$De, !5"

where ni!e" is the ion !electron" density and $Di!e" is the ion

!electron" diffusion time. The diffusion times are function of

the ratio !" /!De"2 and are calculated using experimental re-

sults from Gerber and Gerardo29

or Freiberg and Weaver26

!see Fig. 1". !Only results from Ref. 29 are available for ions

and, consequently, they have been used."The electron temperature relaxes due to energy exchange

in collisions with neutrals. At the initial stage of plasma de-

cay, the electron temperature Te0 is much higher than the

neutral temperature T and will decrease and tend asymptoti-

cally to T. The equation for electron temperature relaxation

is30

dTe

dt= −

Te − 1

$T, !6"

where $T−1=)& /2)me /mivTi

/ len)Te()Te /$T

%, me is the elec-

tron mass, and len is the mean free path of electron-neutral

collision. It can be deduced from Eq. !6" that the electron

cooling time decreases when increasing the pressure.

C. Plasma decay in the presence of dust

In the presence of dust particles in the plasma, the tem-

perature relaxation and diffusion losses could be different

from dust-free plasma. However, no experimental data nor

theoretical estimations of the transition from ambipolar to

free diffusion in dusty plasma exist. Consequently, for our

simulation we will assume that the electron temperature re-

laxation and diffusion processes of ions and electrons are not

significantly affected by the presence of dust particles, allow-

ing us to treat ion and electron diffusion in a similar way to

dust-free plasma. This restricts the validity of our simula-

tions to the case of low dust particle densities. In this case,

the influence of dust particles can be restricted to plasma

absorption losses on the particle surface. Indeed, when im-

mersed in a plasma, a dust particle acquires a net electric

charge due to ions and electrons “falling” on its

surface.10,11,32

In dusty plasmas, ions and electrons captured

by dust particles can be considered as “lost” by the plasma

and thus the charging process of dust particles is also a loss

process.

In the orbital motion limited !OML" approach,10,11,32

the

charge on a dust particle can be calculated from the ion and

electron currents ignoring emission currents,

dQd

dt= Ie + Ii, !7"

where Qd is the charge on the dust particle and Ie!i" is the

electron !ion" current. In a typical laboratory discharge, dust

particles are negatively charged due to higher mobility of the

plasma electrons. The ion current can be expressed as

Ii = 4&rd2niqi* kBTi

2&mi+1/2*1 −

qi)d

kBTi+ , !8"

and the electron current as

Ie = 4&rd2neqe* kBTe

2&me+1/2

exp*−qe)d

kBTe+ , !9"

where )d is the surface potential of the dust particle, rd is the

dust particle radius, and qi!e" is the ion !electron" charge. In

steady state, the charge on a dust particle can be obtained:

Qd = C4&*0rdkBTe

eln, ni

ne*meTe

miTi+1/2- . !10"

For typical argon plasma, me /mi.1.4·10−5 and 1+Te /Ti

+100, and the correction factor C#0.73.33

A dusty afterglow plasma is not a steady case but the

OML approach can be used to obtain the charge on dust

particles. A Fokker–Planck model of dust charging due to the

discreetness of the charge currents !ion and electron" can be

used to obtain dust particle charge distributions. Indeed, the

10−1

100

101

102

103

104

10−1

100

101

102

!

Λ

λDe

"2

Ds

Da

ions (Gerber and Gerardo)

electrons (Freiberg and Weaver)

electrons (Gerber and Gerardo)

ambipolar diffusion

FIG. 1. !Color online" Evolution of

the ratio Ds /Da !s= i ,e" as a function

of !" /!De"2. These data are extracted

from Refs. 29 and 26.

063705-3 Influence of the ambipolar-to-free diffusion… Phys. Plasmas 15, 063705 "2008!


plasma particle absorption time interval as well as the se-

quence in which electrons and ions arrive at the dust particle

surface vary randomly but observe probabilities that depend

on the dust particle potential )d. The probability per unit of

time for absorbing an electron pe!)d" or an ion pi!)d" are

calculated from the OML currents:34

pe = − Ie/e , !11"

pi = Ii/qi. !12"

The ion and electron currents can be calculated using Eqs.

!8" and !9" only when the dust particle charge is negative.

For positively charged dust particles, the ion current is cal-

culated by Eq. !9" with all subscripts changed to i, and elec-

tron current is calculated by Eq. !8" with all subscripts

changed to e. In the simulation, electron and ion densities are

studied in a fluid manner solving numerically Eqs. !4" and

!5" using the same constant time step t f≪$Di!e"0, where

$Di!e"0 is the diffusion time for ions !electrons" at the begin-

ning of the simulation !i.e., the very beginning of the decay-

ing plasma".The contribution of dust particles to plasma losses is

taken into account in the following way.

The charges of Nd dust particles are computed. Knowing

the dust density nd in the plasma, an equivalent volume Veq

to these Nd dust particles can be calculated.

For a dust particle, a time step tpjfor which the probabil-

ity of collecting an ion or an electron of the plasma is

computed:34

tpj= −

ln!1 − R1"

ptot

, !13"

where 0+R1+1 is a random number and ptot= pe+ pi is the

total probability per unit of time to absorb an ion or an elec-

tron.

While / jtpj, t f, an electron or an ion is absorbed during

the time step tpj, and the nature of the absorbed particle is

determined comparing a second random number 0+R2+1

to the ratio pe / ptot. If R2+ pe / ptot, then the collected particle

is an electron; otherwise, it is an ion. The probability pe and

pi are recalculated and a new time step tpj+1is computed.

When / jtpj- t f, one more iteration is applied to the dust

particle. Nevertheless, the probability of collecting an ion or

an electron in the time interval .t= t f −/k=1j−1tpk

is p!.t"=1

−exp!−.tptot" $if tp1- t f, then p!tp1

" is computed%. A random

number 0+R3+1 is then generated and if R3+ p!.t", a

plasma particle !ion or electron" is absorbed. The choice be-

tween ion and electron is decided as previously described.

When all the Nd dust particles have been treated, the

number of absorbed ions and electrons Niabs and Neabs, re-

spectively, by the Nd dust particles during the time step t f is

known and can be transformed using the equivalent volume

Veq into absorbed ion and electron densities niabs and neabs,

respectively, which are subtracted from the ion density ni and

electron density ne before the next iteration of time step t f.

For each iteration, the mean time tp necessary for one

particle to collect a plasma particle !i.e., an ion or an elec-

tron" as well as the ratio rloss=ne!i"diff/ne!i"abs

, where ne!i"diffis

the density of electron !ion" lost by diffusion. The program is

stopped when tp≫$Di ! tp-10$Di in this simulation" when

the charge on dust particles can be considered as frozen due

to a plasma loss time becoming faster than the particle charg-

ing time.

III. RESULTS AND DISCUSSION

In this section, the experimental dust charge distributions

as well as the simulated ones are presented. We then discuss

the validity of our results and role of the transition from

ambipolar to free diffusion.

A. Experimental results

The experimental residual charge distributions have been

obtained using the PKE-Nefedov reactor.1

An upward ther-

mophoretic force was applied to dust particles in order to

counterbalance gravity35

when the discharge is off. To study

particle charges, a sinusoidal voltage produced by a function

generator with amplitude /30 V and frequency of f =1 Hz

was applied to the bottom electrode. The induced low fre-

quency sinusoidal electric field generates dust oscillations if

the dust particles have a residual electric charge. A thin laser

sheet perpendicular to the electrodes illuminates the particles

and the scattered light is recorded at 90° with standard

charge coupled device !CCD" cameras. By superimposition

of video frames, particle trajectories have been obtained. The

coordinates of the particles were measured in each third

frame and the amplitude of the oscillations was determined

from the measured particle positions. Absolute values for the

oscillation amplitude were obtained by scaling the image

pixels to the known size of the field of view.

The charge of a dust particle Qd can then be obtained

from the oscillation amplitude b,25

Qdres=

mdb0)02 + 412/md

2

E0

, !14"

where md is the mass of the dust particle, E0 is the amplitude

of the electric field at the mean height of the dust particle,

0=2&f the angular frequency of the sinusoidal electric field,

and 1 the damping coefficient. The electric field E0 was

taken as the one above the axis of a polarized disk. For a

levitation height of 1.5 cm, the electric field is

E0=6 V /cm.25

The total error on the residual charge mea-

surement !taking into account the error on the dust mass,

dust size, E0, and the damping force" is about #49%. The

sign of the dust particle charge is deduced from the phase of

the dust particle oscillation with respect to the excitation

electric field. A more detailed description of this experiment

can be found in Ref. 25.

Following and measuring oscillations of many dust par-

ticles, the dust particle residual charge distribution in a after-

glow plasma can be constructed. In Fig. 2, dust particle re-

sidual charge distributions are presented for two operating

pressures. It can be seen that for both pressures the mean

residual charge is negative and corresponds to a few elec-



trons !−5e for 1.2 mbar and −3e for 0.4 mbar". In both cases,

the measured residual charge distribution has a tail that ex-

tends into positive residual charge region.

B. Numerical results

The numerical dust particle charge distributions are re-

constructed simulating the charge of Nd=500 dust particles

corresponding to a dust density nd=52104 cm−3. The dust

particle radius was rd=190 nm corresponding to the experi-

mental ones. The initial ion density is ni0=52109 cm−3 and

the initial dust particle charge distribution is computed using

a Cui–Goree algorithm34

and the quasineutrality condition

Zdnd + ne = ni, !15"

where Zd= &Qd /e&. The initial electron density ne0 is deduced

from this calculation. Many iterations of the algorithm are

necessary to obtain the initial dust charge distribution and the

initial electron density: the first iteration assumes ne0=ni0

=52109 cm−3, and for the next iteration ne0 is calculated

using Eq. !15" and the dust particle charges of the first itera-

tion. A new dust particle charge distribution is then com-

puted. This process is performed again and again until ne0

and the dust charge distribution are stabilized. The obtained

dust particle charge distribution is presented in Fig. 3. The

mean charge is Qmean'−952e and the variance '!Qd"'17e. Equation !10" predicts dust particle charges

Qd=−950e, which is in total agreement with our simulation

results.

The decay of a dusty argon plasma is then simulated

using the algorithm previously described for two gas pres-

sures $P=0.4 mbar and P=1.2 mbar !Fig. 4"%. The ion-

neutral mean free path is calculated using the cross section

from Varney31

and the electron-neutral mean free path is cal-

culated using the cross section from Kivel.36

The initial elec-

tron temperature is taken as Te0=3 eV and the ion tempera-

ture is supposed to be equal to the neutral temperature Ti

=Tn=T=0.03 eV. The diffusion length is taken as "=1 cm,

which is approximately the diffusion length of the PKE-

Nefedov reactor in which experiments on residual dust

charge have been performed.24,25

The transition from ambi-

polar to free diffusion is based on either experimental results

from Gerber and Gerardo29

or experimental results from

Freiberg and Weaver26 !the former suggest a slower

transition from ambipolar to free diffusion than the latter; see

Fig. 1".As it can be seen in Fig. 4, the first decrease of the dust

particle charge corresponds to the electron temperature relax-

ation. While electrons and ions diffuse ambipolarly, the

charge remains constant. Finally, when the transition occurs

!after tens of milliseconds", electron and ions densities devi-

ate from each other and the dust charge decreases until it

freezes.

For a pressure P=0.4 mbar !P=0.3 Torr", the simulated

final dust charge distributions !i.e., tp-10$Di" are presented

in Fig. 5. The residual charge is Qdres'−16e when no tran-

sition in the diffusion process is taken into account $Fig.

5!a"%. When using a model based on an abrupt transition

from ambipolar to free diffusion at a Havnes parameter of

PH=0.5, the residual charge is smaller !in absolute value"

−10 −8 −6 −4 −2 0 20

5

10

15

Charge (e)

Num

ber

ofdust

part

icle

s

−10 −5 0 50

10

20

30

40

Charge (e)

Num

ber

ofdust

part

icle

s

P = 0.4 mbar

P = 1.2 mbar

FIG. 2. !Color online" Experimental dust charge distribution. Top: P=0.4 mbar. Bottom: P=1.2 mbar. A Gaussian fit is superimposed to the

experimental distributions.

−1000 −980 −960 −940 −920 −9000

10

20

30

40

50

60

70

Charge (e)

Num

ber

ofdust

part

icle

s

FIG. 3. !Color online" Dust charge distribution for 190 nm radius dust par-

ticles with ni0=52109 cm−3 and nd=52104 cm−3.



Qdres'−13e but still far from experimental value $Fig. 5!b"%.

It should be noted that, typically, the Havnes parameter

reaches 0.5; after 10 ms, it corresponds to a " /!De ratio

close to unity. Using Gerber and Gerardo data !slow transi-

tion, lower curve in Fig. 1", the mean residual charge Qdres

'−13e $see Fig. 5!c"%, whereas it is Qdres'−1e $see Fig.

5!d"% using Freiberg and Weaver data !fast transition, higher

curve in Fig. 1". In the slow transition case, there are no

positive particles observed in the simulated dust particle

charge distribution whereas there are ones in the case of the

fast transition. The residual charge distribution for the fast

transition is similar to the experimental results. !see Fig. 2".For a pressure P=1.2 mbar !P=0.9 Torr", the simulated

final dust particle charge distributions are presented in Fig. 6.

The residual charge is Qdres'−16e when no transition in the

diffusion process is taken into account $Fig. 6!a"%. When us-

ing a model based on an abrupt transition from ambipolar to

free diffusion !at a Havnes parameter of PH=0.5", the re-

sidual charge is smaller !in absolute value" Qdres'−7e but

still far from experimental value $Fig. 6!b"%. A dependence

on the ambipolar to free diffusion transition is again seen

with Qdres'−14e for the slow transition and Qdres

'−2e for

the fast transition. The latest charge distribution is the closest

to experimentally measured distribution.

The results for both pressures are summarized in Table I.

IV. DISCUSSION

We now discuss the results and limits of our model. In

general, we found that the transition from ambipolar to free

diffusion plays a major role in the discharging process of the

dust particle. It is instructive to see how the transition affects

the residual dust charge, in particular, the width of dust

charge distribution. The fact that electrons and ions began to

diffuse independently at an early stage of afterglow changes

the dust charge distribution function drastically.

As we can see from data in Figs. 5!a"–5!c" for the pres-

sure of 0.4 mbar the distributions for ambipolar diffusion,

abrupt, and slow transitions do not differ significantly. Thus,

any difference in diffusion in the late afterglow !low ratio of

" /!De" has little effect on residual charge. In contrast, a

slight difference at an early stage has a large influence on the

100

101

102

103

104

105

105

107

109

Time (µs)

Densi

ty(c

m−

3)

ions

electrons

Zdn

d

b)

100

101

102

103

104

105

0

50

100

Time (µs)

Te/T

i

a)

100

101

102

103

104

105

0

10

20

30

Time (µs)

Dif

fusi

on

tim

e(m

s) ions

electrons

c)

FIG. 4. !Color online" Decay of an argon plasma at P=1.2 mbar with a fast ambipolar-to-free diffusion tran-

sition. !a" Electron temperature relaxation; !b" density

evolution; !c" evolution of diffusion time.



final dust charge distribution and leads to the appearance of

positively charge particles $Fig. 5!d"%. For the higher pres-

sure !1.2 mbar" this effect is not so pronounced, and the

diffusion in the late afterglow plays a noteworthy role. In this

case an abrupt transition directed due to the influence of the

dust charge particle volume effect tends to decrease the re-

sidual dust particle charge $Fig. 6!b"%. It does not, however,

give us the experimentally observed value of residual charge

or the positive tail of the charge distribution. Thus, the re-

sults obtained let us conclude that the four-stage model can

be used only for the rough estimation of residual charge. For

a more accurate calculation, one has to take into account the

actual diffusive rates for the electrons and ions.

We now discuss the validity of the presented model. This

model is valid only for low dust particle density. As the

losses by diffusion are treated as in a dust-free plasma, the

influence of dust particle has to be very small compared to

the total process. Indeed, if losses on dust particles are simi-

lar or bigger than those by diffusion, this last process must be

significantly affected. It has thus been shown that the pres-

ence of a high density of dust particles significantly reduces

the plasma loss time.37

Moreover, if the total dust charge is

not negligible compared that of ions and electrons, the dif-

fusion process must also be modified as the dust particles

will repel the electrons and attract the ions.

Consequently, the influence of dust particles can be

treated independently from the other loss processes only if

the total charge carried by the dust particles is small com-

pared to the charge carried by electrons or ions during the

decay process. Furthermore, losses on dust particle surfaces

must not be the main loss process !i.e., rloss≫1" to allow ion

and electron diffusions to be treated in the same way as in

dust-free plasma. This condition is definitely satisfied for a

discharge plasma but could change during the afterglow so

we have to calculate evolution of the ratio nidiff/niabs

in dis-

−20 −15 −10 −50

20

40

60

80

100

Charge (e)

Num

ber

ofdust

part

icle

s

−20 −18 −16 −14 −12 −10 −8 −60

20

40

60

80

100

Charge (e)

Num

ber

of

dust

part

icle

s

−8 −6 −4 −2 0 2 40

20

40

60

80

100

120

Charge (e)

Num

ber

of

dust

part

icle

s

−25 −20 −15 −100

20

40

60

80

100

120

Charge (e)

Num

ber

ofdust

part

icle

s

a) b)

c) d)

FIG. 5. !Color online" Numerical results for 190 nm radius dust particles with argon pressure P=0.4 mbar !P=0.3 Torr" and nd=52104 cm−3. !a" Ambipolar

diffusion until the end of the decay process. !b" Abrupt transition from ambipolar to free diffusion when PH=0.5. !c" Using data from Gerber and Gerardo for

transition from ambipolar to free diffusion !Ref. 29". !d" Using data from Freiberg and Weaver for transition from ambipolar to free diffusion !Ref. 26".



charge afterglow where nidiffis the ion density lost by diffu-

sion and niabsthe ion density lost by absorption onto the dust

particle surfaces.

It shows that this ratio stay more or less above 5 during

the whole decay process regardless of the conditions of pres-

sure or data used to take into account the transition from

ambipolar to free diffusion !only results using Gerber and

Gerardo data are presented in Fig. 7". It means that during

the plasma decay, losses on dust particles are very small

compared to the losses to the reactor walls and our model for

plasma decay is valid to the end of the afterglow. Further-

more, looking at the ratio &Qd¯ nd /eni& presented in Fig. 8, it

can be seen that, except at the very end of the afterglow !i.e.,

a few milliseconds before the residual charges freeze", the

total dust charge volume is much smaller than the total

charge of ions. The charge onto the dust particles thus rep-

resents a negligible part of the total charge during the major

part of the decay process.

The data presented in Figs. 7 and 8 confirmed that, in

our case, the overall influence of dust particles on the plasma

decay can be neglected and, consequently, the assumption of

dust-free plasma diffusion can be considered as valid for the

whole decay process.

It should be noted that the transition from ambipolar to

free diffusion is not the only process that influence the dust

residual charge. Indeed, it has been shown that in reactive

−22 −20 −18 −16 −14 −12 −100

20

40

60

80

100

120

Charge (e)

Num

ber

ofdust

part

icle

s

−15 −10 −5 00

10

20

30

40

50

60

70

80

Charge (e)

Num

ber

ofdust

part

icle

s

−20 −15 −10 −50

50

100

150

Charge (e)

Nu

mb

er

of

du

stp

art

icle

s

−8 −6 −4 −2 0 2 40

20

40

60

80

100

Charge (e)

Num

ber

ofdust

part

icle

s

a) b)

c) d)

FIG. 6. !Color online" Numerical results for 190 nm radius dust particles with argon pressure P=1.2 mbar !P=0.9 Torr" and nd=52104 cm−3. !a" Ambipolar

diffusion until the end of the decay process. !b" Abrupt transition from ambipolar to free diffusion when PH=0.5. !c" Using data from Gerber and Gerardo for

transition from ambipolar to free diffusion !Ref. 29". !d" Using data from Freiberg and Weaver for transition from ambipolar to free diffusion !Ref. 26".

TABLE I. Dust particle residual charges for two operating pressures.


Experimental results −5e −3e

Ambipolar diffusion −16e −16e

Havnes transition −7e −13e

Slow transition !Ref. 29" −14e −13e

Fast transition !Ref. 26" −2e −1e



plasmas such as C2H4 /Ar plasma,38,39

in which the dust den-

sity can be very high !nd'4.52106 cm−3",39the afterglow

electron density shows an unexpected increase at the very

beginning of the decay process. Berndt et al. attributed this

increase to an electron release by the dust particles.38,39

However, as a such increase has already been observed in

pulsed helium discharges and attributed to a re-ionization

due to metastable-metastable collisions,40,41

and as the pres-

ence of high dust particle density in a argon dilution plasma

enhances the metastable density,42

the increase of the elec-

tron density at the very beginning of the afterglow is still not

clear. In all cases, these “extra electrons” must affect the

charge on dust particles and consequently the residual charge

distributions.

Moreover, it has been shown that the pressure in noble

gases such as argon and neon significantly influences the

diffusion coefficient.43

Indeed, at low pressure the electron

diffusion cooling process, i.e., the fast loss of energetic elec-

trons to the walls of the reactor, which are imperfectly com-

pensated through elastic collisions of the electrons with gas

atoms, resulting in an electron temperature below that of the

gas, causes a reduction of DaP and thus an enhancement of

the ambipolar diffusion time $D="2/Da. As mentioned pre-

viously, the dust particle charges are related to ion and elec-

tron density. Diffusion cooling can also affect dust particle

charge distribution evolution. Nevertheless, the present

model $see Eq. !6"% does not take into account the phenom-

ena of diffusion cooling and its importance has to be inves-

tigated in a future article.

V. CONCLUSION

To conclude, the decharging process of dust particles is

strongly dependent on the transition from ambipolar to free

diffusion as their charges depend on the ratio ni /ne $see Eq.

!10" for equilibrium charge%. In this article, it has been

100

101

102

103

104

0

5

10

15

Time (µs)

nid

if

f/n

ia

bs

raw data

smooth data

a)

100

101

102

103

104

0

5

10

15

Time (µs)

nid

if

f/n

ia

bs

raw data

smooth data

b)

FIG. 7. !Color online" Evolution of nidiff/niabs

using data

from Gerber and Gerardo for transition from ambipolar

to free diffusion !Ref. 29". !a" P=0.4 mbar. !b" P=1.2 mbar.



shown that the character of the transition influences the final

residual dust charge distribution in a strong manner. The

faster the transition occurs !the earlier electrons and ions

start to diffuse separately", the smaller the mean dust particle

residual charge and a positive charge tail can even be ob-

tained. Consequently, it may be possible to use the dust par-

ticle charge distribution as a probe to study the transition

from ambipolar to free diffusion in decaying plasma as this

distribution is strongly dependent on the transition. Further

investigations are being carried on about this possibility.

It has to be noted that the proposed model is valid only

in a limited range of parameters as some important issues

such as the influence of dust particle charge volume, possible

metastable re-ionization, low pressure diffusion cooling,

have been ignored. Consequently, further studies are needed

for a fuller understanding of the decay process in dusty

plasmas.

ACKNOWLEDGMENTS

The authors would like to thank S. Dozias and B. Dumax

for electronic support, J. Mathias for optical support, and Y.

Tessier, C. Cuthbert, and J. Sisovic for experimental support.

The authors would also like to thanks B. W. James for help-

ful discussions. The PKE-Nefedov chamber has been made

available by the Max-Planck-Institute for Extraterrestrial

Physics, Germany, under the funding of DLR/BMBF under

Grant No. 50WM9852. This work was supported by CNES

101

102

103

104

10−2

100

Time (µs)

! ! !

Qdn

d

en

i

! ! !


Fast transition [24]a)

101

102

103

104

105

10−4

10−2

100

102

Time (µs)

! ! !

Qdn

d

en

i

! ! !


Fast treansition [24]b)

FIG. 8. !Color online" Evolution of &Qd¯ nd /eni&. !a" P

=0.4 mbar. !b" P=1.2 mbar.



under Contract No. 793/2000/CNES/8344. This paper is par-

tially supported by the French–Australian integrated research

program !FAST" under Contract No. FR060169.

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Author's personal copy

Physics Letters A 372 (2008) 5336–5339

Contents lists available at ScienceDirect

Physics Letters A

www.elsevier.com/locate/pla

Dust density effect on complex plasma decay

L. Couëdel a,b,∗, A.A. Samarian a, M. Mikikian b, L. Boufendi b

a School of Physics A28, The University of Sydney, NSW 2006, Australiab GREMI (Groupe de Recherches sur l’Énergétique des Milieux Ionisés), CNRS/Université d’Orléans, 14 rue d’Issoudun, 45067 Orléans Cedex 2, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 April 2008

Received in revised form 16 June 2008

Accepted 17 June 2008

Available online 27 June 2008

Communicated by F. Porcelli

PACS:52.27.Lw

Keywords:Complex plasma

Dust

Afterglow

Decay

In this Letter, the influence of dust particles on the plasma losses in a complex plasma afterglow is

studied. It is shown that the dust particles can drastically shorten the plasma loss time by absorption-

recombination onto their surfaces. The dust particle absorption frequency increases with the dust density

but the dependence is not linear for high dust density. Finally, the possible use of dust absorption

frequency measurements as a diagnostic for complex plasmas is mentioned and supported by comparison

to existing experimental data.

2008 Elsevier B.V. All rights reserved.

1. Introduction

Complex plasmas (also called dusty plasmas) are partially

ionised gases in which there are charged dust particles. They

can be natural plasmas (such as near-Earth plasmas, combustion

plasmas, comet tails, planetary rings, interstellar clouds, etc.) or

human-made plasmas (PECVD, etching, fusion reactor, etc.). The

dust particles are charged due to their interactions with the sur-

rounding plasma electrons and ions [1,2]. The presence of this new

charged species leads to interesting phenomena that distinguish

complex plasmas from classic dust-free plasmas [3–5].

In laboratory discharges, the dust particles can be either grown

directly in the plasma chamber (by sputtering [6,7] or using re-

active gases [8,9]) or injected inside the plasma [10,11]. Due to a

higher mobility of the electrons the dust particles are negatively

charged. Consequently, in running discharges, dust particles act as

a sink for the electrons of the plasma [2].

The loss of charged species in a complex plasma afterglow is

of major importance as it determines the charge carried by the

dust particles during the decay process and is fundamental to ex-

plain the dust particle residual charges observed experimentally

in the late afterglow [12,13]. Dust particle residual charge can be

useful (for example, the cleaning of industrial or fusion reactors us-

* Corresponding author at: School of Physics A28, The University of Sydney, NSW

2006, Australia.

E-mail address: [email protected] (L. Couëdel).

ing electrostatic techniques) or harmful (for example, future single

electron devices where residual charge on deposited nanocrystal

would be the origin of malfunction).

In a complex plasma afterglow, the dust particles will increase

the plasma losses as ion and electron absorption-recombination

can occur at their surface. When the dust particle density is high

the particle absorption time τQ decreases. The plasma decay time

τL can thus be drastically shortened. This effect has been observed

by Dimoff and Smy in 1970 [14] but their results were not properly

explained, especially the effect of high dust density on the particle

absorption frequency that was far from their theoretical prediction.

In this Letter, the influence of dust particle density on plasma

losses due to recombination on to dust particle surfaces is dis-

cussed. It is shown that the dust particle absorption frequency

does not grow linearly with the dust density. This effect is visible

when the ratio of the total charge carried by the dust particles to

the charge carried by the ions (Havnes parameter) is close to 1. Fi-

nally, by comparison of the model with existing experimental data,

the possibility of using plasma decay time in a complex plasma as

a diagnostic is discussed.

2. Theoretical approach

When a discharge is switched off, the ions and electrons of the

plasma start to diffuse and the plasma density decreases as fol-

lows:

dn

dt= −

n

τL, (1)

0375-9601/$ – see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.physleta.2008.06.047


L. Couëdel et al. / Physics Letters A 372 (2008) 5336–5339 5337

where n is the density of the plasma and τL the plasma decay

time. In a complex plasma, plasma losses are due to the diffusion

of the ions and the electrons toward the wall of the reactor and by

recombination onto the dust particle surfaces. Consequently, the

plasma decay time τL is:

τ−1L = τ−1

D + τ−1Q , (2)

where τD is the plasma diffusion time and τQ the dust particle

absorption time. As the dust particles are charged due to the fluxes

of ions and electrons flowing to their surfaces, the dust particle

absorption time will depend on the charge (and surface potential)

of the dust particle. Using the orbital motion limited approach the

charge on a dust particle is given by [2]:

dZddt

= Je − J i = πr2d

[

nevTee−ϕ

− nivT i

(

1+Te

T iϕ

)]

, (3)

where Zd is the dust particle charge number, J i(e) is the ion (elec-

tron) flux, rd is the dust particle radius, ni(e) is the ion (electron)

density, vT i(e) is the ion (electron) thermal speed, T i(e) is the ion

(electron) temperature and ϕ is the dimensionless dust surface po-

tential. The latter can be linked do the dust charge number Zd by:

ϕ =Zde

2

4πε0rdkB Te, (4)

where ε0 is the vacuum dielectric permitivity and kB is the Boltz-

mann constant. It sould be noted that the influence of trapped

ions on the dust particle charge is neglected. It has been shown

that when T i # Te and (rd/λD)2 # T i/Te where λD is the De-

bye length, the trapped ions play a significant role in the charging

process of dust particles [15,16] and Eq. (3) has to be modified

appropriately. However, for the conditions we are interested in,

(rd/λD)2 ∼ T i/Te .

In a complex plasma afterglow, recombination on dust particle

surfaces is directly linked to the flux of ion onto the dust particles

as the dust particles are negatively charged in a laboratory plasma

(Q d = −Zde where Q d is the dust particle charge and e is the el-

ementary charge). Consequently, the dust particle absorption time

τQ can be written as:

τ−1Q = πr2dndvT i

(

1+Te

T iϕ

)

. (5)

In a dust-free plasma (ne = ni), the equilibrium surface potential of

an isolated dust particle can be found by solving dZd/dt = 0 and

thus by solving:

(

T i

Te

me

mi

)1/2(

1+Te

T iϕ0

)

exp(ϕ0) = 1. (6)

In a complex plasma, however, the ion density ni and the electron

density ne are different due to the charging of the dust particles.

However, as the plasma is quasi-neutral, the electron density can

be deduced using the neutrality equation:

ne = ni − Zd · nd. (7)

The ratio of the electron density to the ion density can be thus

written as:

neni

= 1−Zdndni

= 1− PH , (8)

where PH = Zdnd/ni is the Havnes parameter. As it can be seen

from Eqs. (7) and (8), for very low dust density, the electron den-

sity is very close to the ion density (ne % ni and PH ∼ 0). Conse-

quently, the surface potential of the dust particle will be the same

as for an isolated dust particle and the dust particle absorption

time τQ 0is inversely proportional to the dust density:

τ−1Q 0

= πr2dndvT i

(

1+Te

T iϕ0

)

. (9)

When the dust density increases, however, the electron and ion

densities start to deviate from each other and the Havnes param-

eter increase. Consequently, solving dZd/dt = 0, the dust surface

potential follows:

(

T i

Te

me

mi

)1/2(

1+Te

T iϕ

)

exp(ϕ) = 1− PH . (10)

This equation has to be solved numerically and the value of the

surface potential explicitly depends on the dust particle density.

This will directly affect the dust particle absorption time. However,

if the dust density is not too high, the deviation of the dimension-

less dust surface potential from the isolated dust particle surface

potential will be small so one can write ϕ = ϕ0 +&ϕ . Using a first

order approximation, Eq. (10) can be rewritten as:

(

T i

Te

me

mi

)1/2(

1+Te

T i(ϕ0 + &ϕ)

)

exp(ϕ0)(1 + &ϕ)

= 1−Zdndni

. (11)

From Eqs. (6) and (4), it can then be deduced that:

&ϕ = −

(

1+

(

Te

T i

me

mi

)1/2

exp(ϕ0) +4πε0rdkB Tend

e2ni

)−1

×4πε0rdkB Te

e2ϕ0ndni

. (12)

In typical laboratory plasma, (Teme/T imi)1/2 exp(ϕ0) ∼ 10−2 and

(4πε0rdkB Tend)/(e2ni) ∼ 10−1 , thus:

&ϕ % −4πε0rdkB Te

e2ϕ0ndni

∼ −Zd0ndni

, (13)

where Zd0 is the charge of an isolated dust particle. The dust par-

ticle absorption time can be rewritten as:

τ−1Q % πr2dndvT i

(

1+Te

T iϕ0

)

− πr2dn2dvT i ·

Te

T i

4πε0rdkB Te

e2ϕ0

ni. (14)

The dust particle absorption time is no longer inversely propor-

tional to the dust density. There is a term in n−2d which tends to

attenuate the effect of the dust particles on plasma losses. Indeed

when the dust density is high the dust surface potential is lower

as is the dust charge number Zd . Consequently the ion flux is re-

duced and the dust particle absorption time is increased compared

to the value it would have if the dust particles can be considered

as isolated.

3. Numerical results

In this section, exact numerical solutions of Eqs. (5) and (10) are

presented and compared to the approximate solution of Eq. (14).

In Fig. 1, the absorption frequency τ−1Q has been calculated

for ni = 108 cm−3 , Te = 0.3 eV (close to experimental values of

Ref. [14]) and a dust particle radius of rd = 10 µm. As it can be

seen in Fig. 1(a), for low dust densities the dust absorption fre-

quency τ−1Q is increasing linearly as expected. On the contrary, for

high dust density the absorption time τ−1Q deviates from linearity.

The exact solution and the first order approximation are in agree-

ment even though some small discrepancies are observed at high

dust density. In Fig. 1(b), the exact solution is presented for dif-

ferent ratio Te/T i . As can be seen, the absorption frequency τ−1Q

increases with increasing Te/T i . Indeed, when the electron temper-

ature is higher than the ion temperature, ion and electron fluxes

are greater at equilibrium (Eq. (3)) resulting in an increased dust


5338 L. Couëdel et al. / Physics Letters A 372 (2008) 5336–5339

(a)

(b)

Fig. 1. (Color online.) Particle absorption frequency as a function of the dust den-

sity. (a) With or without taking into account the dust density effect. (b) Numerical

solution for different T e/T i.

particle absorption frequency. In Ref. [12], it is thus shown that the

electron temperature relaxation in the plasma afterglow of a RF

discharge results in a strong decrease of the dust particle charges.

In Fig. 2, the ratio τ−1Q /τ−1

Q 0as well as the Havnes parameter PH

have been calculated for different dust densities and ion densities.

As can be seen in Fig. 2(a), the deviation from τ−1Q 0

is marked at

high dust density and low ion density. In Fig. 2(b) these conditions

correspond to a Havnes parameter close to 1. For small Havnes

parameters (i.e., small dust density and high ion density), it can

be seen that τ−1Q /τ−1

Q 0∼ 1. This confirms that when dust particles

can be considered as isolated inside the plasma the absorption fre-

quency is directly proportional to the dust density while for high

dust density the charge carried by a dust particle is reduced as the

dust density increases and consequently the dust absorption fre-

quency does not evolve linearly with the dust density.

In a plasma afterglow, the plasma density decreases with time.

Consequently the effect of the dust particles can become more and

more important with time. In order to see this effect, the first mo-

ment of the decay of a complex plasma has been investigated us-

ing Eqs. (1), (2), (5) and (8). The results are presented in Fig. 3. The

plasma diffusion loss is taken into account by choosing a plasma

diffusion frequency τ−1D = 500 Hz (which is a typical value for a

(a)

(b)

Fig. 2. (Color online.) (a) Ratio of the dust particle absorption frequency taking into

account dust density effect τ−1Q over the dust particle absorption frequency not

taking into account dust density effect τ−1Q 0

for different dust density and plasma

density. (b) Havnes parameter PH for different dust density and plasma density.

laboratory discharge with an argon pressure around 1 Torr and

these plasma parameters). As can be seen in Fig. 3(a), the dust

particles enhance plasma losses and the larger the dust density,

the larger the effect on plasma losses. In Fig. 3(b), it can be seen

that at the very beginning of the plasma afterglow, the absorption

frequency is roughly proportional to the dust density. This is due

to the high value of the plasma density at this time which allows

a small Havnes parameter and thus dust particles can be consid-

ered as isolated. However, as the density decreases, the effect of

the dust particles become more and more important and the par-

ticle absorption frequency decreases. Indeed the Havnes parameter

increases with time as the plasma density decreases. Consequently

the dust particle charging is more and more affected by the pres-

ence of the other dust particles. The ion flux as well as the dust

particle charge number decrease leading to a decrease of the dust

absorption frequency. This effect is perfectly visible in Fig. 3(b) for

high dust density.

4. Discussion and conclusion

As we demonstrated in Section 3, the dust density influences

greatly the dust absorption–recombination process in an afterglow

plasma. Thus, the dust absorption frequency does not increase lin-


L. Couëdel et al. / Physics Letters A 372 (2008) 5336–5339 5339

(a)

(b)

Fig. 3. (Color online.) Top: evolution of ion density ni during the first microsec-

onds of a plasma afterglow for different dust density. Bottom: evolution of the dust

particle absorption frequency τ−1Q during the first microseconds of a plasma after-

glow for different dust density. The plasma diffusion frequency is τ−1D = 500 Hz and

Te = T i = 0.3 eV.

early with the density, and for high density it is smaller than the

absorption frequency that would occur if each dust particle could

be considered as isolated in the plasma. Moreover, it was shown in

Sections 2 and 3 that if the dust density is not too high, a first or-

der approximation of the absorption frequency matches very well

with the exact solution. In Eq. (14), there are only a few parame-

ters to know (Te , T i , rd , ϕ0 and ni). Consequently, it is possible to

use the measurement of the absorption frequency in the afterglow

of a complex plasma in parallel with other diagnostics to deduce

some important parameters of the plasma.

For example, in Ref. [14], the dust particle absorption frequency

has been measured in the afterglow of a linear pulsed discharge.

The argon pressure was 1.9 Torr and the density of dust var-

ied from 0 to 104 cm−3 . The dust particles had a mean radius

rd % 15 µm. It is reported that the electron temperature was

roughly equal to the ion temperature (Te ∼ T i) [14]. Consequently,

the influence of trapped ions on the dust particle charge can be

neglected. In Fig. 4, experimental results from Ref. [14] are fitted

using Eqs. (9) and (14). The fit gives T i % 0.042 eV, (Te/T i)ϕ0 =

4.4 and thus assuming T i ! Te ! 5T i (depending when the mea-

surement have been performed in the plasma afterglow), it gives

107 cm−3 ! ni ! 5 × 107 cm−3 . The range for the ion density and

the ion temperature are much lower than those that were sup-

posed in Ref. [14] but it could be due to the fact that their mea-

surements were performed quite late in the plasma afterglow and

temperature relaxation needed to be taken into account.

Fig. 4. (Color online.) Fit of experimental results of Ref. [14] using the first order

approximation (Eq. (14)).

To conclude, the influence of the dust density on plasma losses

has been studied. It has been shown that the presence of dust par-

ticles can drastically shorten the plasma loss time and that the

increase of the dust absorption frequency does not depend lin-

early on the dust particle density. Finally, by comparing our results

with existing experimental data, the possible use of absorption fre-

quency measurement as a diagnostic for complex plasma is men-

tioned. Comparison with existing experimental data [14] supports

this idea.

Acknowledgements

The authors would also like to thanks B.W. James for help-

ful discussions. This Letter is partially supported by the French–

Australian integrated research program (FAST) under contract

No. FR060169.

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nanoparticle formation and dynamics in a complex (dusty) plasma

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